Is there a name for this type of percussion setup?

The kind Sheila E. played? It’s two snare drums on a stand, one or more cymbals, and a cowbell or two. I’ve also seen the same setup used by a member of the Miami Sound Machine.

If I wanted one of these setups, could just go to the music store and say, “I want a _______” or would I have to piece it together myself?

Generically, you’re talking about Afro-Cuban percussion. This site even has a “Sheila E” line of drums (along with a variety of instruments from other traditions):

http://www.tocapercussion.com/

They were invented to call up the spirits (Santos, Loa) & they still do the job.

Maybe she puts snares on a timbale stand? IANAD.

Actually, yeah, if you google ‘timbale stand’ you will see your setup.

Timbales. They’re not snares, per se.

Thanks.