$\newcommand\Ring{\mathrm{Ring}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Spec{Spec}$Here is how you can define the notion of an Zarsiski-open subfunctor starting only with the Zariski topos $\mathcal Z$ of sheaves on $\Ring^\text{op}$ and its canonical internal ring object $\mathbb A$. The input data is $(\mathcal Z,\mathbb A)$, and nothing more.
In fact, allow me to describe a process which produces from any Grothendieck-topos $\mathcal E$ together with an internal (commutative) ring object $R$ in $\mathcal E$ a category of "schemes", such that when you plug $(\mathcal Z,\mathbb A)$ into it, it gives you the actual category of schemes. I will assume that the pair $(\mathcal E,R)$ satisfies a minimal amount of the axioms of synthetic differential geometry. In particular I will assume that $R$ is an internal field in the sense that the following is true. $$\mathcal E\models
\forall x:R. \, \neg (x=0)\to \exists y:R. x y=1$$
And I will assume that $0\neq 1$ in $R$. Both of these conditions are satisfied in the algebraic model $(\mathcal Z,\mathbb A$). As you can see, I use categorical logic.
Step 1. Single out the affine spaces: The internal ring object structure of $R$ gives each hom-set $\Hom_\mathcal E(X,R)$ the structure of a ring which we denote by $\mathcal O(X)$. Pulling back functions preserves the ring structure, and so we get a contravariant functor $\mathcal O:\mathcal E\to \Ring$. We say that an object $X$ in $\mathcal E$ is affine if and only if for every other object $Y$ it holds that $$\Hom_\mathcal E(Y,X)\to \Hom_{\Ring}(\mathcal OX,\mathcal OY)$$
is a bijection. In other words a morphism into $X$ is determined by and can be defined through how it pulls back the functions on $X$. This is a definition which doesn't make much sense for the standard models of SDG (because there $\mathcal O(X)$ has actually the structure of a $C^\infty$-ring), but in the case $(\mathcal Z,\mathbb A)$ it gives us precisely the representables inside $\mathcal Z$.
Step 2. Define the Zariski open subobjects of affine spaces. Fix an affine space $X$. For each function $f:X\to R$, let us declare the subobject $$D(f)=\{x:X|\neg f(x) = 0\}$$ of $X$ to be open. They are the basic open subobjects. Since we like the open subobjects of $X$ to form a lattice, let us also declare arbitrary joins (defined through colimits in $\mathcal E$) of such $D(f)$ to be open in $X$. Now we know what the open subspaces of affine spaces are.
Lemma to step 2. If $X$ is an affine space, then the poset of open subobjects of $X$ has not only arbitrary joins, but also finite meets. Furthermore, the join and meet operations in the lattice of opens agree with those of the surrounding category.
Proof. Let us check that the intersection $D(f)\wedge D(g)$ is $D(fg)$, where the multiplication of $f$ and $g$ is defined by postcomposition with the ring-structure multiplication map of $R$. The other statements follows automatically. To make this work, we actually need that $R$ is a field from the internal perspective. Because of that $D(f) = \{x:X|\, \ulcorner f(x)\text{ is invertible}\urcorner\}$ and since a product of elements in a ring is invertible if and only if both components are (and the proof of that algebra fact is intuitionistically valid), we see that indeed $D(f)\wedge D(g) = D(fg)$. $\square$
Remark. This is a definition which uses categorical logic. You can translate it into a categorical definition (which involves an equalizer) via the semantic described in Jacobs book, or in the case of a sheaf topos you can use the forcing semantic described in Palmgren - Constructive sheaf semantics to translate it in a concrete situation. What you will find out (after translation) is that in the case $(\mathcal Z,\mathbb A)$ and for an affine scheme $\Spec A$ the subobject $D(f)$ is precisely $\Spec A[1/f]\to \Spec A$. If $I$ is an ideal in $A$ and you form the join $\bigvee_{f\in I} D(f)$ in $\mathcal Z$, then you will see that the resulting subobject is precisely the subobject $D(I)$ associated to $I$ as it is described in Demazure & Gabriel's book.
Step 3. Define a topology on arbitrary objects. Given any object $Y$ in $\mathcal Z$, we equip it with the strongest topology such that all maps from affine spaces into $Y$ stay continuous. This means that a subobject $U$ is open in $Y$ if and only if for every affine space $X$ and every morphism $\phi:X\to Y$ the preimage $\phi^{-1}U$ (defined through a pullback) is open in $X$.
Lemma to step 3. We need of course check that the new definition doesn't introduce any new open subobjects on affine spaces. It does not.
Proof. Assume $U$ is a subobject of an affine space $X$ such that the pullback to every other affine space is open. Then in particular $id_X^{-1}U = U$ is open in $X$, and we see that $U$ is already open in $X$. Conversely take any st-indexed join $\bigvee_iD(f_i)$ of basic open subobjects $D(f_i)$ in $X$, and let $\phi:Y\to X$ be a test map from an affine space. Then $$\phi^{-1}(\bigvee_iD(f_i)) = \bigvee_i \phi^{-1}D(f_i)= \bigvee_i D(f_i\circ \phi) $$
for categorical reasons, and we see that the pullback is open in $Y$. $\square$
Final step. Define the category of schemes. A scheme is an object $X$ which can be covered by some collection of affine open subobjects $U_i$. Here covered means that $\bigvee_iU_i=X$. Alternatively the induced map $\sqcup_i U_i \to X$ is an epimorphism in $\mathcal E$.
If you apply that process to the stating data $(\mathcal Z, \mathbb A)$, then you get the category of schemes. Crucially you need not know what the representables are or what the defining site of $\mathcal Z$ is, you only need the internal line $\mathbb A$.
Edit. Here are some texts which explain the translation of internal statements in categorical logic to external meaningful statements.
The link above is the paper Constructive Sheaf Semantics by Erik Palmgreen. It has a nice list of translation rules on the fourth page, and I always use it in praxis.
There are two texts by Andrew Pitts which are really beginner friendly, concise and clear. They are called Categorical Logic and Notes on Categorical Logic and can be found online.
The ultimate reference is the first half of Bart Jacobs' book Categorical Logic and Type Theory.
I think this should be enough, but if this post happend to get you interested in categorical logic, feel free to text me. I have collected a huge number of lecture notes and books, which I am happy to share. :)
Even though the forcing semantic of categorical logic is site dependent, the standard semantic (as explained e.g. in Jacobs' book) is not. It depends only on the category and its limits and colimits. The external statement one gets through the forcing semantic is always equivalent to the translation one gets through the categorical semantic, even though they might look different. In that sense the forcing semantic is a useful tool, when a defining site is available, but the justification for categorical logic comes from the categorical semantic. I am just adding this so that no one thinks that I am sneaking in a site-dependency through the forcing semantic!