Is ZF + AD consistent with: For every real $r$, every true $Σ^V_2(r)$ statement has a $Δ^V_2(r)$ example?
DC is provable in ZF + every true $Σ^V_2$ statement has a $Δ^V_2$ example (i.e. witness). Also, in ZF, for $Σ^V_2(r)$ statements, an $\mathrm{OD}(r)$ example implies existence of a $Δ^V_2(r)$ example.
$Σ^V_2$ is essentially the broadest possible class here (if we want $\mathrm{OD}(r)$ examples) since under AD, there is a non-OD real number, and being non-OD is $Π^V_2$. We also cannot extend $r$ to countable ordinals, even if the examples are real numbers rather than arbitrary sets. On the other hand, under $\text{AD}^+$ (and thus in $L(ℝ)$ under AD), for every real $r$, every $Σ^2_1(r)$ statement has a $Δ^2_1(r)$ example. But under $\text{AD}^+ + V=L(P(ℝ))$, OD reals have a $Δ^2_1$ well-ordering, so $Σ^2_1$ is as far as we can go there.
Without the parameter $r$, I think the answer is yes: Start with $L(ℝ)$, and maintaining $V=HOD(ℝ)$, and using countably closed forcing that does not add sets of reals, encode a real $x$ for which some desired example is $Δ^V_2(x)$. Repeat without upsetting the previous examples and encodings until (after a countable number of steps) for all true $Σ^V_2$ statements, an example is encoded. For encoding $x$, we can use the continuum function, and to restore $V=HOD(ℝ)$, we can encode the generic set into the continuum function, and iterate that $ω$ times. (I am not sure how to prove that no sets of reals are added, and I will upvote an answer that completes the argument, though the accept checkmark is for the top question.)
With the parameter $r$, the existence of $Δ^V_2(r)$ examples implies the axiom of choice for sets parameterized by reals, and hence, under AD, $\text{AD}_ℝ$ + "$Θ$ is regular". The axiom of choice for sets parameterized by reals holds because otherwise (by assumption) it would have a $Δ^V_2$ counterexample $A⊂ℝ⨯X$, with every nonempty $A[r]$ containing a $Δ^V_2(r)$ element, but this allows uniformization of $A$.
ZF + $\text{AD}_ℝ$ + "$Θ$ is regular" + $V=L(P(ℝ))$ (this implies DC) proves that there is $s∈Θ^ω$ such that for every real $r$, every true $Σ^V_2(r)$ statement has a $Δ^V_2(s,r)$ example. Proof: Using $V=L(P(ℝ))$, we get an example that is $Δ^V_2$ definable from a set of reals (dependent on $r$). Using "$Θ$ is regular", the required Wadge ranks are bounded (so for some $B⊆ℝ$ independent of $r$, we get an example definable from $B$ and a real). Using uniformization, there is $A⊆ℝ$ (independent of $r$), such that we get $Δ^V_2(A,r)$ examples. Finally, using the scale property, $A$ can be encoded into $s$.
We can encode $s$ into the model by forcing (I think without adding new sets of reals), but the problem is that new $Σ^V_2(r)$ statements will become true, and iterating an uncountable number of times may destroy AD. Instead, we would need to do something subtle: Examples for $Σ^V_2(r)$ statements would need to be encoded but the decoding would need to use $r$ as a key to work.
