Gimp is a slur for someone with a disability or physical deformity. It has also come to mean someone incapable or incompetent. The ADHS finds usages dating from the 1920s, many of them referring to someone with a bad leg who therefore limps. However, there is an earlier (1877) adjectival usage meaning inferior, second-rate.
The expression on the gimp seems unique to Gone in 60 seconds, where it appears here as well:
MEMPHIS: You know where the others are?
Donny frowns... Looks closely at Memphis ... Then:
DONNY ASTRICKY: Most of 'em are gone. The Dyar Boys are doing a nickel at Chino; Henry Santoro and Frankie Fish are moving weight in Florida; Bill Doolin was killed in Denver... Atley Jackson's on the gimp and runnin' errands for Calitri;
So "on the gimp" refers to retired car thieves who no longer have the skills to boost cars and one Atley Jackson who apparently was once a car thief and is now reduced to the status of errand boy.
I think it's fair to infer from its usage in the film and as documented elsewhere that it means someone who's lost his skills and cannot be considered competent.