Lots of free activities show up on Eventbrite with plenty for children (but this does depend where you live). Local community hubs / centres / churches and youth groups tend to put on summer holiday events. Is there a local cubs / scouts group?
What about volunteering? My friend's children are volunteering at the local foodbank this summer. The adults there will adore your son and they might also have children in tow.
He might be too shy for sports clubs, but perhaps you could consider a specific activity that he could take lessons in, including horse riding, which is a shared experience but not competitive? Or chess, which is competitive but tends to attract the quieter, brainier type.
Another friend takes her children to volunteer at the local nature reserve. It suits her children's temperaments, which are similar to your son's. Again, the adults there love them and children go along too, plus they learn so much valuable information about nature, team work, responsibility...
There might be a music group he could join, for example a ukelele group which would be fun and he would be learning a great skill. Music is a good skill for life. Give him guitar lessons. Everybody loves a good guitarist.
Set him up for his future, not just his present, by considering something educational that involves others, including adults, who will treat him with respect. That will be invaluable to his sense of self-worth.
Best of luck - your son sounds adorable (keep telling him how fab he is and how lucky you are) 