ODO on form
The relevant entry is 7c; none of the others really fit the context:
7 [mass noun] the state of a sports player or team with regard to their current standard of play:
they are one of the best teams around on current form
- details of previous performances by a racehorse or greyhound:
an interested bystander studying the form
- a person’s mood and state of health:
she seemed to be on good form
- British informal a criminal record:
they both had form
In OED it’s moved down to 16c:
c. slang. (Without preceding article.) A ‘police record’; a criminal conviction.
In this case it doesn't actually mean “a criminal record”; it means “a history of criminality” or “a history of conflict against each other”.