Let for every $\epsilon>0$ there is $\delta>0$ such that for every homeomorphism $f:X\to X$ with $d_{C^0}(f, id)<\delta$, we have $d(a, f^n(a))<\epsilon$ for all $n\in\mathbb{Z}$.
What can say about $a\in X$?
Is it true that $a$ is an isolated point for $X$?
I don't know what you mean by $d_{C^0}$, but regardless of what that metric is you cannot conclude that a is an isolated point. Indeed, consider a subspace of the euclidean plane shaped like a cross (for example the union of the x- and y-axis) and let $a$ be the center of that cross. Then any homeomorphism from the cross onto itself must fix $a$. This is because for any other point on the cross there is a sufficiently small open ball around that point such that when removing a point from that ball the resulting set has two connected components, whereas when you take any open neighbourhood of $a$ and remove $a$ the resulting set has four or more connected components. Since $a$ is fixed by any homeomorphisms it trivially satisfies the property in your question, yet $a$ is not isolated.