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1. $C$ is algebraic
2. For all $c \in C$, $F(c)$ is algebraic.
3. For all $c \xrightarrow{f} c'$ in $C$, the functor $f^*:F(c') \to F(c)$ has a left adjoint $f_! \dashv f^*$.
4. For all $c \xrightarrow{f} c'$ in $C$, the functor $f^*:F(c') \to F(c)$ preserves sifted colimits.

To see that the set $\{(p,q_p)\}$ generates $\int F$ under sifted colimits, take $(c,a) \in \int F$. Then there is a sifted category $I$ such that $\varinjlim\limits_{i \in I} p_i=c$, in particular there are structure morphisms $\iota_i:p_i \to c$. For each $i \in I$, we have that $\iota_i^*a$ is a sifted colimit of some $q_{p_i,j}$ or equivalently, $a$ is the sifted colimit of some $(\iota_i)_!q_{p_i,j}$, so if we take the colimit over $(p_i,q_{p_i,j})$, we get $(c,a)$.

1. $C$ is algebraic
2. For all $c \in C$, $F(c)$ is algebraic.
3. For all $c \xrightarrow{f} c'$ in $C$, the functor $f^*:F(c') \to F(c)$ has a left adjoint $f_! \dashv f^*$.
4. For all $c \xrightarrow{f} c'$ in $C$, the functor $f^*:F(c') \to F(c)$ preserves sifted colimits.
5. For all $c \xrightarrow{f} c'$ in $C$, the functor $f^*:F(c') \to F(c)$ is conservative.

Conditions 4. and 5. imply that for each $p \xrightarrow{f} c$ the set $\{f_!q_p\mid q_p \in P_{F(p)}\}$ generates $F(c)$ under sifted colimits. Now take $(c,a) \in \int F$. Then we can find a filtered $I$ category and a diagram $I \to C, i \mapsto p_i$ such that $c=\varinjlim\limits_{i \in I} p_i$ with structure morpisms $\iota_i:p_i \to c$. Then for all $i \in I$, we can find a filtered category $J_i$ and a diagram $J_i \to F(c), j_i \mapsto (\iota_i)_!q_{p_i,j_i}$ such that $\varinjlim\limits_{j_i \in J_i}(\iota_i)_!q_{p_i,j_i}=a$. Now we get that $\varinjlim\limits_{i \in I} \varinjlim\limits_{j_i \in J_i}(p_i,q_{p_i,j_i})=(c,a)$

1. $C$ is algebraic
2. For all $c \in C$, $F(c)$ is algebraic.
3. For all $c \xrightarrow{f} c'$ in $C$, the functor $f^*:F(c') \to F(c)$ has a left adjoint $f_! \dashv f^*$.

Condition $3.$ implies that $\int F \to C$ is in fact a Grothendieck bifibration. Because it is an opfibration and the base category and the pointwise images are cocomplete, the total category is cocomplete and colimits may be computed as follows: for a diagram $I \to \int F, i \mapsto (c_i,a_i)$, first take the colimit $\varinjlim c_i$ in $C$, denote the structure morphisms of that as $\iota_i$ and then take the colimit in $F(\varinjlim c_i)$ of $(\iota_i)_!a_i$, then the pair $(\varinjlim c_i, \varinjlim (\iota_i)_! a_i) \in \int F$ is the desired colimit. Now we can choose for $C$ a set of compact projectives $P_C$ that generates $C$ under sifted colimits. And then for each $p \in P_C$, we may choose a set of compact projectives $P_{F(p)}$ in $F(p)$ that generate it under sifted colimits, then the set $\{(p,q_p) \mid p \in P_C, q_p \in P_{F(p)}\}$ is a set of compact projectives for $\int F$ that generates it under sifted colimits, which one can check because colimits and morphisms both have an explicit reasonable description. Hence $\int F$ is algebraic.

1. $C$ is algebraic
2. For all $c \in C$, $F(c)$ is algebraic.
3. For all $c \xrightarrow{f} c'$ in $C$, the functor $f^*:F(c') \to F(c)$ has a left adjoint $f_! \dashv f^*$.
4. For all $c \xrightarrow{f} c'$ in $C$, the functor $f^*:F(c') \to F(c)$ preserves sifted colimits.

Condition $3.$ implies that $\int F \to C$ is in fact a Grothendieck bifibration. Because it is an opfibration and the base category and the pointwise images are cocomplete, the total category is cocomplete and colimits may be computed as follows: for a diagram $I \to \int F, i \mapsto (c_i,a_i)$, first take the colimit $\varinjlim c_i$ in $C$, denote the structure morphisms of that as $\iota_i$ and then take the colimit in $F(\varinjlim c_i)$ of $(\iota_i)_!a_i$, then the pair $(\varinjlim c_i, \varinjlim (\iota_i)_! a_i) \in \int F$ is the desired colimit. Now we can choose for $C$ a set of compact projectives $P_C$ that generates $C$ under sifted colimits. And then for each $p \in P_C$, we may choose a set of compact projectives $P_{F(p)}$ in $F(p)$ that generate it under sifted colimits, then the claim is that the set $\{(p,q_p) \mid p \in P_C, q_p \in P_{F(p)}\}$ is a set of compact projectives for $\int F$ that generates it under sifted colimits.

Fix such a pair $(p,q_p)$. Let $I$ be a sifted category and consider a diagram $I \to \int F, i \mapsto (c_i,a_i)$. As described above, we have the colimit of the diagram given by $(\varinjlim c_i, \varinjlim (\iota_i)_!a_i)$. We have:

$$ \mathrm{Hom}_{\int F}((p,q_p),(\varinjlim c_i, \varinjlim (\iota_i)_!a_i))=\{(f,\alpha) \mid f \in \mathrm{Hom}_C(p,\varinjlim c_i),\alpha \in \mathrm{Hom}_{F(p)}(q_p,f^*(\varinjlim (\iota_i)_!a_i))\}$$ Now we can use condition 4. and that $q_p$ is compact projective, so we get that this is $$\{(f,\alpha) \mid f \in \mathrm{Hom}_C(p,\varinjlim c_i), \varinjlim \mathrm{Hom}_{F(p)}(q_p, f^*((\iota_i)_!a_i))\}$$ $p$ is also compact projective, so this is (using that $I$ is sifted) $$\varinjlim \{(f,\alpha) \mid f \in \mathrm{Hom}_C(p,c_i),\alpha \in \mathrm{Hom}_{F(p)}(q_p,f^*a_i)\}=\varinjlim \mathrm{Hom}_{\int F}((p,q_p),(c_i,a_i))$$

To see that the set $\{(p,q_p)\}$ generates $\int F$ under sifted colimits, take $(c,a) \in \int F$. Then there is a sifted category $I$ such that $\varinjlim\limits_{i \in I} p_i=c$, in particular there are structure morphisms $\iota_i:p_i \to c$. For each $i \in I$, we have that $\iota_i^*a$ is a sifted colimit of some $q_{p_i,j}$ or equivalently, $a$ is the sifted colimit of some $(\iota_i)_!q_{p_i,j}$, so if we take the colimit over $(p_i,q_{p_i,j})$, we get $(c,a)$.

@Z.M pointed out in the comments that a useful framework to look into is animation. If we want a $1$-category to to have an animation, the natural question to ask is if is an algebraic category, which here means being cocomplete and generated under sifted colimits by a set of compact projective objects.

As I have written in the OP, I realized after asking the question that $\mathbf{GrpMod}$, or to be precise the fibered category $\mathbf{GrpMod} \to \mathbf{Grp}$ is actually the Grothendieck construction of the pseudofunctor $G \mapsto \Bbb Z[G]\textrm{-}\mathbf{Mod}$.

Thus to answer the question, I will provide a sufficient criterion for the Grothendieck construction of a pseudofunctor to be algebraic which implies that $\mathbf{GrpMod}$ is algebraic.

So let $C$ be a category and let $F:C^{op} \to \mathbf{Cat}$ a pseudo-functor. Suppose that:

1. $C$ is algebraic
2. For all $c \in C$, $F(c)$ is algebraic.
3. For all $c \xrightarrow{f} c'$ in $C$, the functor $f^*:F(c') \to F(c)$ has a left adjoint $f_! \dashv f^*$.

Then $\int F$ is algebraic.

Condition $3.$ implies that $\int F \to C$ is in fact a Grothendieck bifibration. Because it is an opfibration and the base category and the pointwise images are cocomplete, the total category is cocomplete and colimits may be computed as follows: for a diagram $I \to \int F, i \mapsto (c_i,a_i)$, first take the colimit $\varinjlim c_i := c$ in $C$, denote the structure morphisms of that as $\iota_i:c_i \to c$ and then take the colimit in $F(c)$ of $(\iota_i)_!a_i$, then the pair $(\varinjlim c_i, \varinjlim (\iota_i)_! a_i) \in \int F$ is the desired colimit. Now we can choose for $C$ a set of compact projectives $P_C$ that generates $C$ under sifted colimits. And then for each $p \in P_C$, we may choose a set of compact projectives $P_{F(p)}$ in $F(p)$, then the set $\{(p,q_p) \mid p \in P_C, q_p \in P_{F(p)}\}$ is a set of compact projectives for $\int F$, which one can check because colimits and morphisms both have an explicit reasonable description. Hence $\int F$ is algebraic.

Therefore we get the animation $\mathbf{Ani}(\mathbf{GrpMod})$. The coinvariants functor $(-)_G:\mathbf{GrpMod} \to \mathbf{Ab}$ is cocontinuous and hence induces a functor $H_{\mathrm{Ani}}:\mathbf{Ani}(\mathbf{GrpMod}) \to \mathbf{Ani}(\mathbf{Ab})$ which deserves to be called animated group homology. One can recover the usual $n$-th homology functor as the composition $$\mathbf{GrpMod} \to \mathbf{Ani}(\mathbf{GrpMod}) \xrightarrow{H_{\mathrm{Ani}}} \mathbf{Ani}(\mathbf{Ab})=\mathbf{D}^-_{\geq 0}(\mathbf{Ab})\xrightarrow{\pi_n} \mathbf{Ab}$$

This answers the question on whether one can do a form of homological algebra with $\mathbf{GrpMod}$: through animation we have injected it with homotopical/derived information and obtained a form of a derived category.

Intuitively, I would expect that, like $\mathbf{GrpMod}$ is the collection of the categories of $\Bbb Z[G]\textrm{-}\mathbf{Mod}$ for all groups $G$ "glued together" along all the morphisms in the category $\mathbf{Grp}$, the $\infty$-category $\mathbf{Ani}(\mathbf{GrpMod})$ is the collection of all derived $\infty$-categories $\mathbf{D}^-_{\geq 0}(\Bbb Z[G]\textrm{-}\mathbf{Mod})$ glued together over $\mathbf{Ani}(\mathbf{Grp})$.

To maybe make this precise one can note that $\mathbf{GrpMod} \to \mathbf{Grp}$ is cocontinuous, so we obtain an animated functor $\mathbf{Ani}(\mathbf{GrpMod}) \to \mathbf{Ani}(\mathbf{Grp})$. I'm not sure if it's a reasonable conjecture, but one could certainly raise the question whether this is a Cartesian (op?) fibration, and whether the fiber over an object $G$ in the image of $\mathbf{Grp} \to \mathbf{Ani}(\mathbf{Grp})$ is of the form $\mathbf{Ani}(\Bbb Z[G]\textrm{-}\mathbf{Mod})$. If that holds, the other fibers should then be the analogue of the derived category of $\Bbb Z[G]$-modules for animated groups.

@Z.M pointed out in the comments that a useful framework to look into is animation. If we want a $1$-category to to have an animation, the natural question to ask is if is an algebraic category, which here means being cocomplete and generated under sifted colimits by a set of compact projective objects.

As I have written in the OP, I realized after asking the question that $\mathbf{GrpMod}$, or to be precise the fibered category $\mathbf{GrpMod} \to \mathbf{Grp}$ is actually the Grothendieck construction of the pseudofunctor $G \mapsto \Bbb Z[G]\textrm{-}\mathbf{Mod}$.

Thus to answer the question, I will provide a sufficient criterion for the Grothendieck construction of a pseudofunctor to be algebraic which implies that $\mathbf{GrpMod}$ is algebraic.

So let $C$ be a category and let $F:C^{op} \to \mathbf{Cat}$ a pseudo-functor. Suppose that:

1. $C$ is algebraic
2. For all $c \in C$, $F(c)$ is algebraic.
3. For all $c \xrightarrow{f} c'$ in $C$, the functor $f^*:F(c') \to F(c)$ has a left adjoint $f_! \dashv f^*$.

Then $\int F$ is algebraic.

Condition $3.$ implies that $\int F \to C$ is in fact a Grothendieck bifibration. Because it is an opfibration and the base category and the pointwise images are cocomplete, the total category is cocomplete and colimits may be computed as follows: for a diagram $I \to \int F, i \mapsto (c_i,a_i)$, first take the colimit $\varinjlim c_i$ in $C$, denote the structure morphisms of that as $\iota_i$ and then take the colimit in $F(\varinjlim c_i)$ of $(\iota_i)_!a_i$, then the pair $(\varinjlim c_i, \varinjlim (\iota_i)_! a_i) \in \int F$ is the desired colimit. Now we can choose for $C$ a set of compact projectives $P_C$ that generates $C$ under sifted colimits. And then for each $p \in P_C$, we may choose a set of compact projectives $P_{F(p)}$ in $F(p)$ that generate it under sifted colimits, then the set $\{(p,q_p) \mid p \in P_C, q_p \in P_{F(p)}\}$ is a set of compact projectives for $\int F$ that generates it under sifted colimits, which one can check because colimits and morphisms both have an explicit reasonable description. Hence $\int F$ is algebraic.

Therefore we get the animation $\mathbf{Ani}(\mathbf{GrpMod})$. The coinvariants functor $(-)_G:\mathbf{GrpMod} \to \mathbf{Ab}$ is cocontinuous and hence induces a functor $H_{\mathrm{Ani}}:\mathbf{Ani}(\mathbf{GrpMod}) \to \mathbf{Ani}(\mathbf{Ab})$ which deserves to be called animated group homology. One can recover the usual $n$-th homology functor as the composition $$\mathbf{GrpMod} \to \mathbf{Ani}(\mathbf{GrpMod}) \xrightarrow{H_{\mathrm{Ani}}} \mathbf{Ani}(\mathbf{Ab})=\mathbf{D}^-_{\geq 0}(\mathbf{Ab})\xrightarrow{\pi_n} \mathbf{Ab}$$

This answers the question on whether one can do a form of homological algebra with $\mathbf{GrpMod}$: through animation we have injected it with homotopical/derived information and obtained a form of a derived category.

Intuitively, I would expect that, like $\mathbf{GrpMod}$ is the collection of the categories of $\Bbb Z[G]\textrm{-}\mathbf{Mod}$ for all groups $G$ "glued together" along all the morphisms in the category $\mathbf{Grp}$, the $\infty$-category $\mathbf{Ani}(\mathbf{GrpMod})$ is the collection of all derived $\infty$-categories $\mathbf{D}^-_{\geq 0}(\Bbb Z[G]\textrm{-}\mathbf{Mod})$ glued together over $\mathbf{Ani}(\mathbf{Grp})$.

To maybe make this precise one can note that $\mathbf{GrpMod} \to \mathbf{Grp}$ is cocontinuous, so we obtain an animated functor $\mathbf{Ani}(\mathbf{GrpMod}) \to \mathbf{Ani}(\mathbf{Grp})$. I'm not sure if it's a reasonable conjecture, but one could certainly raise the question whether this is a Cartesian (op?) fibration, and whether the fiber over an object $G$ in the image of $\mathbf{Grp} \to \mathbf{Ani}(\mathbf{Grp})$ is of the form $\mathbf{Ani}(\Bbb Z[G]\textrm{-}\mathbf{Mod})$. If that holds, the other fibers should then be the analogue of the derived category of $\Bbb Z[G]$-modules for animated groups.