I do not understand the question.
The answer to this is probably either trivial for one reason or trivial for another reason.
If your group has a mascot that is not violating any rights, your group effectively has the rights to that mascot, if there are any rights to have at all. And subsequently your group can give permission to use that character in other games, or permission is not needed.
If the character falls under rights you do not have, you cannot give permission. But you might ask for permission. Like, if there was an artist that created the mascot, there might be permission for your group to use it as a mascot, but not yet permission for everyone to use the character in a game.
But the way your worded the topic and text might also indicate that you wonder if the topic as such would be allowed. To which the trivial answer would be, why would it be forbidden? A jams topic can be anything. The games would have to be within the terms of service and not violate laws, of course.