Is there a name for that specific art style that most furry youtube animators had in the late 2000s/early 2010s? It was almost kemono, but with more of a toony western influence. It was especially popular with sparklecat characters.

Examples:
post #3527017 post #4083234

It's just western kids trying to draw chibi art & manga inspired art in general, and does not really have specific name. Also for the record it's not a youtube thing, it was mostly associated with deviantart, and the youtube content was just the deviantart kids expanding to animation. (source: I was one of those kids)

Chibi? At the very least it's heavily influenced by chibi characters in manga. The floating orb hands and simplified/exaggerated expressions were directly pulled from manga as, at the time at least, having small cutaway reaction shots of highly simplified, blob-like versions of the characters was a popular form of comic relief.
This carried over to deviantart, the Warriors fandoms, and the Neopets art/comic/roleplay community. I don't really know if anyone ever gave it a name beyond 'chibi' or 'super deformed' but it is definitely a distinct genre and era of furry art.

The nostalgia xD
I don't think that syle have a name besides Chibi drawn by begginer artists in 2010

I'd probably call it something like "early YouTube furry chibi animation style"