What is the source for the custom that Yemenite Jews people count Sefira in Aramaic? and why do they do it?
Otzar Ta'amei Haminhagim (cf.) explains that it is a remnant of a time when the spoken language was Aramaic. Since the primary purpose of s'fira is the keeping track of days it is preferable to count in a language that enables the counter to keep track - i.e. a language the counter understands.
They preserve the Babylonian custom, where the native tongue of most people was Aramaic. The earliest attestation of saying the sefira in Aramaic of which I am aware is in the siddur of R. Saadia Gaon (see here, bottom of the page).
ופשוט שמותר לספור בכל לשון ודוקא בלשון שמבין ואם אינו מבין לשון הקודש וספר בלשון הקודש לא יצא דהא לא ידע מאי קאמר ואין זה ספירה כן נראה לי.
And obviously, it is allowed to count in all language, only language which he understands; if he doesn't understand Holy Language (Hebrew) and has counted in Holy Language, he doesn't fulfill the Mitsva, because this is not a count.
The source of this discussion may be in Mishna and Gemara Massechet Sota, (Mishna 7, 1) we can understand from the Tosfot Yom Tov that the intended effect of some verbal mitsvot contains understanding. For counting it is easy to understand.
What if someone said and understand and someone ho doe's not understand wants to fulfill by hearing him? The Berayita in Menachot 65b taught: "וספרתם לכם, שתהא ספירה לכל אחד ואחד" that each one need to make the count. So, the count of the few people who understand is not relevant for the "non-understanders". But the Bet Yosef reported from the Teshuvat Harashba that one may fulfill the mitsva through the count of the Shaliach Tsibur. Magen Avraham explains that by the rule of Shomea Keone (who heard is as if repeat what he heard). We can ask if this can help when the person who heard does not understand what is said, but who speak understant what he says. I am pretty sure that the answer is negative.