Does there exist a ring $R$ such that there are autoequivalences of the category of left or right $R$-modules that are not naturally isomorphic to the obvious ones induced by automorphisms of $R$?
Morita equivalences always induce at least an isomorphism between the centers, and "self-Morita equivalences" are no exception. In particular, they must induce an automorphism of the center, and any ring that answers the question must be noncommutative.
Hint: Try to find a ring $R$ such that there is an idempotent $e$ in $R$ other than 1 for which $ReR=R$ and the corner ring $eRe$ is isomorphic to $R$.
$\mathbb{C}\times M_2(\mathbb{C})$.
Or any other ring of the form $A\times B$ where $A$ and $B$ are nonisomorphic but Morita equivalent.