Is there a good general definition of "sheaves with values in a category"? Let $\mathcal{A}$ be a category.
There is a common definition of "sheaves with values in $\mathcal{A}$", which is what one obtains by taking the Grothendieck-style definition of "sheaf of sets" (i.e. in terms of presheaves satisfying a certain limit condition with respect to all covering sieves) and blithely replacing $\textbf{Set}$ with $\mathcal{A}$.
In my view, this is a bad definition if we do not assume $\mathcal{A}$ is sufficiently nice – say, locally finitely presentable.
When $\mathcal{A}$ is locally finitely presentable, we obtain various properties I consider to be desiderata for a "good" definition of "sheaves with values in $\mathcal{A}$", namely:

*

*The properties of limits and colimits in the category of sheaves on a general site with values in $\mathcal{A}$ are "similar" to those of $\mathcal{A}$ itself.
(I am being vague here because even when $\mathcal{A}$ is locally finitely presentable, the category of sheaves with values in $\mathcal{A}$ may not be locally finitely presentable – this already happens for $\mathcal{A} = \textbf{Set}$.)

*The category of sheaves on a site $(\mathcal{C}, J)$ with values in $\mathcal{A}$ is (pseudo)functorial in $(\mathcal{C}, J)$ with respect to morphisms of sites.
(By "morphism of sites" I mean the notion that contravariantly induces  geometric morphisms.)

*The construction respects Morita equivalence of sites, i.e. factors through the (bi)category of Grothendieck toposes.

*The construction respects "good" (bi)colimits in the (bi)category of Grothendieck toposes, i.e. sends them to (bi)limits of categories.
(I don't know what "good" should mean here, but at minimum it should include coproducts.
When $\mathcal{A}$ is locally finitely presentable, there is a classifying topos, so in fact the construction respects all (bi)colimits.)

*The category of sheaves on the point with values in $\mathcal{A}$ is canonically equivalent to $\mathcal{A}$.

*The category of sheaves on the Sierpiński space with values in $\mathcal{A}$ is canonically equivalent to the arrow category of $\mathcal{A}$.

Question.
What is a (the?) "good" definition of "sheaves with values in $\mathcal{A}$"?

*

*... when $\mathcal{A}$ is finitely accessible, not necessarily cocomplete, e.g. the category of Kan complexes, or the category of divisible abelian groups?

*... when $\mathcal{A}$ is an abelian category, not necessarily accessible, e.g. the category of finite abelian groups, or the category of finitely generated abelian groups?

*... when $\mathcal{A}$ is a Grothendieck abelian category, not necessarily locally finitely presentable?

Perhaps something like right Kan extension along the inclusion of the (bi)category of presheaf toposes into the (bi)category of Grothendieck toposes might work – but the existence of toposes with no points suggests it may not – but it would be nice to have a somewhat more concrete description.

There is a temptation to strengthen desideratum 6 to require that the category of presheaves on a (small) category $\mathcal{C}$ be equivalent to the category of functors $\mathcal{C}^\textrm{op} \to \mathcal{A}$, but this does not appear to be a good idea.
As Simon Henry remarks, if $\mathcal{C}^\textrm{op}$ is filtered, then the unique functor $\mathcal{C} \to \mathbf{1}$ is a morphism of sites corresponding to a geometric morphism that has a right adjoint (i.e. the inverse image functor itself has a left adjoint that preserves finite limits), so  contravariant (bi)functoriality with respect to geometric morphisms forces the induced $\mathcal{A} \to [\mathcal{C}^\textrm{op}, \mathcal{A}]$ to have a left adjoint, i.e. $\mathcal{A}$ must have colimits of shape $\mathcal{C}^\textrm{op}$.
This is the same argument that shows that the category of points of a topos must have filtered colimits.
Since I am looking for a construction where $\mathcal{A}$ is not necessarily the category of models of a geometric theory, I conclude that I cannot require the category of presheaves on $\mathcal{C}$ to be $[\mathcal{C}^\textrm{op}, \mathcal{A}]$.
 A: This is a very complicated question.
Categories of Sheaves.
Let me start from something that you evidently know (given the hidden references in your question). Bourceux et al. have worked on  defining sheaves of something over a site $(C,J)$.


*

*Borceux, Sheaves of algebras for a commutative theory, Ann. Soc. Sci. Bruxelles Sér. I 95 (1981), no. 1, 3–19

*Borceux and Kelly, On locales of localizations, J. Pure Appl. Algebra, Volume 46, Issue 1, 1987, Pages 1-34.

*Borceux and Veit, On the Left Exactness of Orthogonal Reflections
J. Pure Appl. Algebra, 49 (1987), pp. 33-42.

*Borceux, Subobject Classifier for Algebraic Structures. Subobject classifier for algebraic structures
J. Algebra, 112 (1988), pp. 306-314.

*Veit, Sheaves, localizations, and unstable extensions: Some counterexamples. J. Pure Appl. Algebra, Volume 140, Issue 2, July 1991, Pages 370-391.

*Borceux and Quinteiro. A theory of enriched sheaves. Cahiers de Topologie et Géométrie Différentielle Catégoriques 37.2 (1996): 145-162.


As you mention, this theory works at its best when $\mathcal{V}$ is a regular locally finitely presentable monoidal closed category (these are the working hypotheses of the last reference in the list above). This is partially satisfactory, as in these instances we recover a nice correspondence between topologies and lexreflective localizations, and thus we can maintain the intuition for the theory of sites and lex-reflectors. On the other hand, this theory does not even recover the notion of sheaves over a topos $\mathsf{Sh}((C,J), \mathcal{E})$, as many topoi are not locally finitely presentable.
An idea. I never checked, but I always believed that most of these results can be recovered when $\mathcal{V}$ is a cocomplete precontinuous category (in the sense of Adamek, Rosicky and Vitale, see On Algebraically Exact Categories and Essential Localizations of Varieties) with a dense generator (and of course monoidal closed). This assumptions would recover the topos case and also the case of Grothendieck categories, where precontinuity is known as (AB5). (This framework would also meet all your desiderata). Of course this idea is not entirely satisfying for you, as it would not encompass those finitely accessible categories that are not cocomplete, but at least provides a good framework to study sheaves over a cocomplete precontinuous monoidal closed category with a dense generator (which, again, include topoi and Grothendieck categories).
Grothendieck topoi. The first dishonest way to answer your question is to say that you are looking for a way to describe lex-reflective $\mathcal{V}$-categories. This is the over-formalist point of view of who thinks that lex-reflectivity of categories of sheaves is not a theorem in the theory of topoi, it is a very intrinsic characterization and should be taken as a definition. I am not sure that I support this idea.
Anyway, if one wants to follow this path, there is the beautiful paper by Garner and Lack.

Garner and Lack, Lex Colimits. J. Pure Appl. Algebra. Volume 216, Issue 6, June 2012, Pages 1372-1396.

As the assumptions on $\mathcal{V}$ are very mild in this case, one cannot expect to recover a good theory of sites, and thus the very intuition of sheaf is a bit lost. Still, depending on your religious belief, this could be a starting point.
Caveat 1. I think that this tentative solution unveils the first problem of your question. If you do not choose a flavour of problems that you want to solve, or attack with this notion of sheaf, it's hard to come with a correct definition which does not rely on a specific point of view.
Caveat 2. Even the formalists that are fashinated by this approach, should be warned by the evidence. In the case of Grothendieck categories the correct notion of morphism is not that of left exact cocontinuous functors, as discussed in the very introduction of a paper of Ramos Gonzalez and myself.

Di Liberti and Ramos Gonzalez, Exponentiable Grothendieck categories in flat algebraic Geometry. arXiv:2103.07876.

So one should be very careful with getting carried by this point of view.
Elementary topoi and finite abelian groups. I was always fascinated by a very natural way to produce elementary topoi. The category of functors $\mathsf{FinSet}^C$ is an elementary topos, and when $C$ is a finite category it is even a Grothendieck topos with respect to the Grothendieck universe of finite sets. This case is very similar to that of finite abelian groups. So, are you honestly changing the notion of sheaf, or just dishonestly changing the notion of size? I think that this is a question to think about before forcing a definition that we might not need.
Kan complexes.You listed Kan complexes among finitely accessible categories that are not cocomplete, but is this the correct point of view on them? Kan compleses are indeed cocomplete with respect to the relevant notion of colimit, and if you want to see this cocompleteness, you should take sheaves over simplicial sets and study model topoi in the sense Rezk.

Rezk, Toposes and homotopy toposes.

All in all. I do not have a good answer to your question, but I have discussed a couple of remarks that I hope will thicken the debate. But I have a very informal question for you, what are the defining and conceptual features of the notion of sheaf that you want to model? Personally, I find your list of desiderata nor defining nor conceptual.
A: The naive definition of sheaves is very well behaved if you look at functoriality in the $f_*$ direction: Of course, you are going to need to assume that $\mathcal{A}$ has all limits as the definition of $\mathcal{A}$-valued sheaves involves arbitrary limit.
If you want to restrict to category that have for example finite limits, you are going to have to restrict to sites that have only have finite cover, and to geometric morphisms that satisfies some finiteness conditions.
Once you assume that $\mathcal{A}$ has finite limits, it works pretty much without any problems. In fact it has little to do with Grothendieck topologies and works well for arbitrary "limit sketches".
Definition: (maybe not completely standard terminology) By a "limit sketches" I mean a small category $\mathcal{C}$ together with $S$ a set of maps in the category  $\widehat{C}$ of presheaves of sets on $\widehat{C}$.
A site is a special case with $S$ the set of covering sieves.
Given $\mathcal{E}$ a category with colimits, any functor $f:C \to \mathcal{E}$ induces an adjunction $f_! \dashv f^*$ where $f^*$ is the nerve functor $\mathcal{E} \to \widehat{C}$ and $f_!:\widehat{C} \to \mathcal{E} $ is the pointwise left kan extention of $f$.
Definition: A $(C,S)$-comodel in $\mathcal{E}$ is a functor $f:C \to \mathcal{E}$ such that $f_!$ sends all maps in $S$ to isomorphism in $\mathcal{E}$, or equivalently such that for all $X \in \mathcal{E}$, $f^* E$ is orthogonal to all maps in $S$.
Now, if $\mathcal{A}$ is a category with all limits, then a $(C,S)$-model is $\mathcal{A}$ is a a $(C,S)$-comodel in $\mathcal{A}^{op}$.
In particular, the category of $(C,S)$-model in $Set$ is the full subcategory of presheaf on $C$ that are orthogonal to all maps in $S$. I'm denoting this category by $Set(C,S)$
Propostion: For all category $\mathcal{E}$ with all colimits, the category of $(C,S)$-comodel in $\mathcal{E}$ is equivalent to the category of left adjoint functors (equivalently, colimit preserving functors) $Set(C,S) \to \mathcal{E}$.
Indeed functor $C \to \mathcal{E}$ corresponds to left adjoint functors $\widehat{C} \to \mathcal{E}$ and by definition $(C,S)$-model corresponds exactly to the ones that inverts all maps in $S$, but by classical manipulation, this is the same as left adjoints functors $Set(C,S) \to \mathcal{E}$. As $Set(C,S)$ is locally presentable, this is also the same as colimit preverving functor.
You get your functoriality requirement:
Corollary: The category of $(C,S)$-(co)model in a category with all (co)limit is functorial on all functor preserving all (co)limits.
In particular, it is functorial on site and Geometric morphisms.
Indeed, for comodel it it justs the functor represented by $Set(C,S)$ in the category of all colimits preserving functor between cocomplete category. It follows for model by duality.
A: In my view, the correct notion of "sheaf of Xs" is "internal X in the topos (or $\infty$-topos) of sheaves of sets (or spaces)".  (I mentioned this previously on MO here.)  Since sheaves of sets are a limit theory, if X is also defined by a limit theory (i.e. the category of Xs is locally presentable), then by commutation of limits this is the same as a sheaf of Xs in the naive sense.  But for other values of X it gives different answers.   In fact, the answer it gives may depend on exactly how the theory of X is presented; but that's reasonable becaues sometimes there is more than one correct notion of "sheaf of Xs" (equivalently, there is more than one version of X in the internal constructive logic of a topos).  For instance:

*

*If X = fields, there are discrete fields, Heyting fields, and residue fields.  I think discrete fields are the one that corresponds to viewing fields as models of a limit-colimit sketch (i.e. as an accessible category), but the others are often more useful (e.g. Heyting fields include the sheaf of continuous real-valued functions on a topological space).

*The case of X = Kan complexes has already been mentioned in other answers.  Although in general once you're talking about homotopy theory, it's better to incorporate the homotopy theory into the ambient $\infty$-topos and work with stacks.

*If X = finite abelian groups, there are different notions of finite object in a topos.

*If X = topological spaces, you can internalize that directly, but often more useful is to internalize the notion of locale -- for instance, a "sheaf of locales" on a sufficiently nice topological space $Y$ is equivalent to a space over $Y$.

*If X = local rings, written as a geometric theory, this definition gives you the generally accepted definition of "sheaf of local rings", i.e. a sheaf of rings whose stalks are local.

This definition of "sheaf of Xs" satisfies your criteria (3) and (5).  It also satisfies your criterion (1) in as strong a way as I think could be expected: the category of internal Xs in a topos behaves exactly like the ordinary category of Xs, as long as the latter is interpreted using constructive logic.  And it satisfies your criteria (2), (4), and (6) if the theory of Xs is geometric, hence has a classifying topos -- which I think is the most general situation in which one can expect these properties to hold.
(Note, by the way, that your criterion (6), as well as the stronger version referring to all presheaf toposes, is a special case of your (4), since presheaves on $C$ are the Cat-enriched copower of the terminal topos by $C$ in the bicategory of toposes.)
