I hit Google and found a few farms in California that grow them. You could probably just call them and ask them who distributes their crops.
https://fig-tree-farms.com/index.html
https://www.pearsonranch.com/buy-finger-limes-online/
It also sounds like the harvest is during the autumn/winter, based on the notes on the first one, so they'd be out-of-season now, so I doubt that you could get them grown in California anyway until late in the year. I have no idea whether they ship well, but if so, you're probably stuck looking for imports if you want them prior to that.
If you have an appropriate place to grow them, you might save the seeds from one if you can get one and plant them and have a tree yourself. It sounds like they do grow from seed (not like, say, the Hass avocado).
EDIT: One more trick. It's not as nice an interface for actually shopping as, say, Amazon or the like, but if you want to find something truly esoteric that you cannot find anywhere else, Google Shopping indexes a ludicrous number of retailer sites. My first search there found:
...and a bunch more hits.