As already mentioned, they are still used to some extent in Okinawan and Ainu languages. There is however also a very limited usage in standard Japanese.
The distinction in sound between ゐ-い and ゑ-え has not been around since the end of the Kamakura period (1333) and in 1946 the Japanese government released the 現代仮名遣｛げんだいかなづか｝い in an attempt to standardize the writing. Here they decided to remove ゐ ゑ as they are no more necessary, considering their pronunciation has merged with their counterparts. Why they decided to keep を however I do not know, possibly because it is only used as a particle. Considering the long time since they were actually being pronounced differently, it's doubtful that the distinction has been able to survive in any Japanese dialect and in the scientific literature I've read there's been no mention of its actual usage anywhere in modern times.
As for its usage in writing, it's mainly in historical names as well as loan words that entered Japanese quite early, such as ヰスキー. The loan words in normal texts have since been replaced, but you can still encounter them, mainly in brands, e.g. ニッカヰスキー. The beer Yebisu also sometimes uses ヱビス, named after the Shinto god 恵比須, which however comes from that Ye could be spelt using both エ and ヱ after their respective sounds had already merged.
It is still allowed to use these special kana in names in the population register and while it's uncommon to see people being given those spellings nowadays, they were frequently used before the war, so there are still many elderly people with them.