Stoic philosophy seems to share a lot with Buddhist wisdom. Since Zeno of Citium (generally credited as the founder of Stoicism) is said to have lived a couple hundred years after Siddhartha Gautama, it seems possible that some cross pollination of ideas may have influenced the development of Stoic philosphy. Is there any evidence (other than the similarities in philosophy) that this happened?
It's hard to answer these kind of things with a negative but I'm going to be bold and do that with caveats. I haven't heard of Stoicism itself being directly influenced by Buddhism but.....
There is evidence that Indian Philosophy was influencing Greek philosophy
The founder of skepticism Pyrrho travelled with Alexander the Great into India and was said to have studied philosophy there. One of his subsequent central concerns was ataraxia (freedom from worry) which to my mind seems to have parallels with the doctrine of liberation as espoused by Buddhism and other Eastern traditions
A specific example of an advocate of east/west cross pollination of ideas is Thomas McEvilley in his book the Shape of Ancient Thought. He contends that modern thought is actually influence by Eastern and Western traditions and there was a rich cross pollination of ideas. He goes right back to the Pre-Socratic and says that with the exception of Thales, these philosophers could (would???) have been influenced by Eastern though as mediated through the Persian Empire.
Looking at it the other way, the transfer of ideas certainly happened the other way around. The Questions of King Milander detail a dialog between a Buddhist Monk and a Indo-Greek king thought to have been Menander. So it's very evident that a dialog was happening between east and west.
So generally the conclusions could be that there was no direct influnce but if the Pre-Socratics and/or the Skeptics were being infuenced by Eastern thought then Stocism itself would have been influenced by those philosophies. The influences doesn't need to be direct to by present and relevant.
We might not/never know
We probably lack direct evidence for this kind of transfer from the Stoics in particular. Philosophers such as Aristole do name check their influences in their texts (and then go on to say why their own ideas were so much better). I'm not aware of any direct referencing to Buddhist or other Eastern traditions in Greek texts.
That said the transmission of ancient texts is clearly very incomplete. I find it fascinating that we have lost much (all??) of the work of Chrysippus one of the main Greek Stoic thinkers. I heard it said that if his works had have come down to us then Chrysippus would have been a philosopher know to us in the same way as Plotinus or even Plato and Aristole.
So that said the eveidence of direct influence could well be there. Given that writings of major thinkers in the Stoic tradition are lost it can't be said what else is in there.
It might not just be the Stoics
I'm personally not convinced that the Stoics would be the best candidates for a heavy Buddhist influence. There is a lot in Stoicism such as their logic and physics that I would struggle to fit into this framework. To my (naive) mind the Skeptics have more similarities or even the Neoplatonic thinkers or even the Pythagoreans. I could even make a (bad) argument for Epicuricans if we consider the central place of spiritual community - though that's really pushing it to an almost crazy extent.
The presence of Greek kingdoms in (what are now) Afganistan, Pakistan and India after the time of Alexander, implies that the evolution of (some schools of) Buddhism was to some extent (small or large depending on what scholar you ask) influenced by Greek thinking. So, it might be that Stoicism influenced the evolution of Buddhism (but then there must likely have been some common ground in terms of thinking already for this to have happened).
Well there was the Indo-Greek kingdom and many of the Greeks living there were Buddhist. This kingdom existed before stoic philosophy came about, so there is a possibility. https://en.wikipedia.org/wiki/Greco-Buddhism