While vamos in Spanish does mean "let's go!", it took on another meaning in America, and later on a different spelling: vamoose
: to depart quickly
In the 1820s and '30s, the American Southwest was rough-and-tumble
territory—the true Wild West. English-speaking cowboys, Texas Rangers,
and gold prospectors regularly rubbed elbows with Spanish-speaking
vaqueros in the local saloons, and a certain amount of linguistic
intermixing was inevitable. One Spanish term that caught on with
English speakers was vamos, which means "let's go." Cowpokes and dudes
alike adopted the word, at first using a range of spellings and
pronunciations that varied considerably in their proximity to the
original Spanish form. But when the dust settled, the version most
American English speakers were using was vamoose.
Given the time, Mark Twain probably still used the original spelling, but with the new meaning, so what he wrote means:
"If I was running this shop I'd make him say something, some time or
other, or [exit] the ranch."
I don't know the context, but presumably the character being talked about doesn't talk much and the speaker doesn't like that and wants to make him leave the ranch.