My kids do/have done karate. It's a great sport for exercise and, actually, for social stuff. There is some social interaction: Sparring and other partner activities, for example. But the group is so inclusive and accommodating that everyone seems to be met right where they are socially.

The only thing that might be challenging (and seems common in martial arts) is belt test to move up levels. Often those are conducted individually or in small groups in front of a larger group. But the studios with which I'm familiar (between my kids and my friends' kids, that covers a few martial arts studios) also offer private testing for a small additional fee.

Maybe look at a couple boxing/martial arts studios in your community to get a feel for which might be a good fit? Some are pretty gung-ho about the sport; others are more laid back. The smaller family-run studios seems to be a little more relaxed and accommodating.