I don’t know anything about The Checkmates, who put out one 45 on Flic Records 786 in the early or mid 1960s. Tim Garbocz wrote “Do It” with vocal by Mickey Stillson. There’s no publishing info listed.
Crypt Records put “Do It” on the 2015 compilation Ho-Dad Hootenanny Too!, which I recommend buying if you like this crude, rockin’ garage sound. The instrumental flip is only a minute and a half long and has not been reissued before.
The initals MW on the label (MW 544/5) indicate Midwest Record Pressing in Chicago pressed this single. Could be they were an Illinois group.
One source shows this band connected to the Checkmates from Johnson City, Tennessee who recorded “Talk to Me” / “Cindy” on Champ Records 2009 in August 1967. That may be possible, but I doubt it – the band would have had to add an organ player and the vocalist sounds different. The Checkmates who recorded “Do It” were a pre-British Invasion group in their style. It would have been a leap for them to get to the sound on “Talk to Me”.