First film with Robinson and Bogie together. Maybe one interior shot of a Cadillac but lots of 1936 Buicks including a light colored coupe with skirts. And Joan Blondell had a really neat Packard. There was one shot of Edward G. in a convertible coupe, but they never showed it below the belt line. (You can't "pause" TCM when it's live to make sure of all the makes.)
According to Internet Movie Car Database its a 33 355C and a 36 series 60.
https://www.imcdb.org/movie_27407-Bullets-or-Ballots.html
This has come up on other forums. Warner Brothers pictures prominently featured Buicks. Think about Casablanca in the airport scene at the end or Bette Davis and her 1940 Buick wagon in Now, Voyager or Jack Benny in George Washington Slept Here. Pretty much every prominently shown car in a Warner Brothers picture of that time was a Buick. It would be interesting to know what kind of a deal they had.
Today it seems to me that most cars shown in modern 1920s to 40s period films are Model A Fords. I think there are 2 reasons: one is that that there are so many of them still extant; the second is that it's simplest for the "Transportation Captain" (the guy in the production company in charge of cars) to just call one of the many Model A clubs and ask for the cars needed.