Apparently the US Supreme Court decided that 27 years ago in Bernal. There the Court ruled Texas could not require a notary to be a citizen.
Nonetheless, the official notary web sites for 5 states still contain a US citizenship requirement:
Kansas
Missouri (see page 16)
Oklahoma
Rhode Island
South Carolina
Presumably a resident alien could sue and get a quick summary judgment allowing an application.
Still, it's been almost 30 years since the Supreme Court made its ruling!
Image courtesy of Salvatore Vuono