Normally new users are not allowed to up-vote until they gain a bit of reputation by receiving up-votes or accepts from the community of a site. But this has a weak point that is being exploited:

A number of new users post, answer, and accept each others' off-topic questions before we can close them (in less than 5 min). Accepting each others' answers they gain enough reputation to start up-vote each others' posts.

We are closing them and the moderators clean up afterwards but this keeps happening over and over again and becoming frustrating to the extent that I think a systematic solution might be required. Is there a system to prevent this kind of circumvention of the reputation requirements using accepts by new users?

One possible solution might be to change the system so accepts from users who do not have enough reputation to up-vote do not give any reputation (at least until the answer gets an up-vote).