Yes. By the late 1920s, black-and-white film could be developed privately the same day, and in a hurry it could be done within hours rather than days.

The practical method would be much like black-and-white processing today: load the exposed film into a tank or daylight-loading tank in darkness, then run it through developer, a brief stop bath, and fixer, followed by washing and drying. Daylight tanks already existed, so once the film was loaded, the rest could be done with the lights on.

Typical negative development was on the order of about 10 minutes, with only seconds in stop bath and a few minutes in fixer. The slowest part for actually viewing the result would often be washing and drying the negatives enough to inspect or print them.

For 35mm specifically, note that factory preloaded 135 cassettes did not appear until 1934, so in the late 1920s the film would typically be loaded into reusable cassettes from bulk stock. That makes the setup slightly less convenient, but not unrealistic.

So for your story, a private investigator could plausibly improvise a dark space, use a tank process, and get usable negatives the same day—potentially quite quickly if only a first look was needed.