I am looking for an algorithm that given a PDA $M$ decides whether there exists $w \in L(M)$ for which there exists a decomposition $w = uvxyz$ that satisfies $|vy| \geq 1$ and $uv^ixy^iz \in L(M)$ for every $i \geq 0$.
In a context-free language $L$, by the Pumping Lemma, there exists a $p > 0$ such that every $w \in L$ such that $|w| \ge p$ admits a decomposition $w = uvxyz$ such that $uv^ixy^iz \in L$ for all $i \geq 0$.
If $L$ is infinite, it has words of length $\ge p$, and you can pick any of them. If $L$ is finite then no word satisfies the condition (this is not a contradiction with the pumping lemma, because the pumping lemma gives a length $p$ which is more than the length of any of the words in $L$).
It is decidable if a context-free grammar generates an infinite language (see e.g. Is it decidable whether a given context free grammar generates an infinite number of strings?).
Therefore the algorithm is
On input $\langle M \rangle$, where $M$ is a PDA
- Convert $\langle M \rangle$ to the description of an equivalent CFG $\langle G \rangle $
- Check if $L(G)$ is infinite
- If yes, then answer yes else answer no