I’ve heard variations on those syllables. They resemble the words “gee” and “haw”, which, as voice commands for draft animals and sled dogs, can be traced back to at least the early 1700s.

“Yee-haw” is also very similar to “Hee Haw”, the title of the 1969 TV show that presented “cornpone” humor interspersed with country music. The term “hee-haw” has long been an onomatopoetic representation of a mule or donkey’s bray.

However, it seems to me that the term is probably an institutionalization of various exclamatory syllabifications that express unbridled enthusiasm and joy, whether it be for music, a rodeo performance, or a powerful jolt of moonshine.

In that usage I think it could be American country and western culture.

The OED lists the first usage in print as the New York Times in 1929. That’s not to say that it wasn’t used prior to that just that they can’t find it written down. It also notes that it’s the Mikasuki word for wolf so it could be that theyn shouted wolf while jumping on their horse to get away / round up the cattle etc and cowboys had seen them do it and started copying.

According to the Google NGram Viewer “Yee-haw” first showed up in literature around early 1870 and usage became popular for fifteen years until disappearing in 1885, then appearing again in 1896 to 1903. It didn’t reappear again until 1983 and has been with us ever since. It’s usage in literature began a sudden, rapid rise in popularity in 1993—about the same time as common usage of the internet began—and peaked in 2004 at an all time high to a record frequency of 2,242% from 1993 frequencies.