I can show that the claim is true if $A$ is defined over $\mathbb F_{p^\infty}$, the algebraic closure of the field with $p$ elements. This proof will unfortunately not generalize to arbitrary algebraically closed fields (not even such of characteristic $p$), nor will it bound the exponent $n$ such that $\Omega^n \cong \rm id$ in any useful way. It does however show that it suffices to know that each simple module is periodic in order to conclude that $\Omega^n\cong \rm id$ for some $n$ (again, of course, only for algebras over $\mathbb F_{p^\infty}$).

Fix $n\in\mathbb N$ such that
$\Omega^{n}(S) \cong S$ for any simple $A$-module $S$.

All but the last paragraph is the same as Theorem 2.5 in *K. Erdmann and A. Skowronski. Periodic algebras. Trends in Representation Theory and Related Topics. European Math.
Soc., Zurich, 2008.* (you can view the relevant parts of this on google books).

First choose a projective cover $P$ of $_A A_A$ (i.e., $P$ is a projective $A$-$A$-bimodule). Then choose an $A$-$A$-bimodule $X$ as the kernel of the epimorphism from $P$ to $A$, i.e. we get a s.e.q. of $A$-$A$-bimdoules
$$
0 \longrightarrow X \longrightarrow P \longrightarrow A \longrightarrow 0
$$
As a sequence of left or right $A$-modules this sequence is split, hence $X$ is projective as a left and as a right $A$-module. This implies that tensoring the above sequence with any (left or right) $A$-module will again yield an exact sequence of left $A$-modules, with projective middle term (since $P\otimes_A M$ is projective for any $A$-module $M$). This shows that
$-\otimes_A X$ is isomorphic to $\Omega(-)$ on the stable module category. Now $-\otimes_AX$ is a stable auto-equivalence of Morita-type, and so is $-\otimes_A X^{\otimes n}$. The latter sends simple modules to themselves, and therefore lifts to a Morita auto-equivalence, w.l.o.g. induced by the $A$-$A$-bimodule $Y$ (the fact that stable equivalences of Morita type which send simple modules to simple modules lift to Morita equivalences is usually attributed to Linckelmann). That is, $-\otimes_A Y \cong \Omega^n(-)$ on the stable module category. In particular $S\otimes_A Y \cong S$ for all simple $A$-modules $S$. But any Morita-autoequivalence which sends simple modules to themselves is induced by an automorphism $\alpha$ of $A$, i.e. $Y$ is isomorphic to the twisted $A$-$A$-bimodule $_{id} A_{\alpha}$.

Now comes the ugly part which doesn't work over arbitrary fields: $\alpha: A \longrightarrow A$ must have finite order, since every non-zero element in $\mathbb F_{p^\infty}$ is a root of unity and therefore every invertible matrix over $\mathbb F_{p^\infty}$ has finite order ( ${\rm Aut}(A) \leq {\rm GL}(A)$, and ${\rm GL}(A)$ is a torsion group). Hence there is some $m\in \mathbb N$ such that $Y^{\otimes m} \cong {_AA_A}$. But then
$$\Omega^{m\cdot n}(-)\cong -\otimes_A Y^{\otimes m} \cong -\otimes_A {_{id} A _{\alpha^m}} \cong \rm id$$