At the playground, you or your in-laws or some other well-known adult are there for him. At this new environment, he's being left with people who are strangers to him. Kids who are outgoing on the playground can be outgoing because they have the security of knowing their adults are nearby; they may well be very different in a new environment. Remember, he's barely 2. This does not mean he won't love school later, so don't fear that or he'll pick up on your fears.
I'm assuming from what you wrote that he's not in a daycare center but at home with the grandparents all day. So this is a bigger change for him than you may have thought. You didn't say if this "preschool/camp" is half-day, full-day etc. I'd check with the people running it and if they're professionals who do this all the time and know this age group, they may have ideas and you could try a bit more, hang around a few minutes until he's engaged with a toy or new friend before you leave, etc. But if they're not professional preschool teachers/caregivers, I'd reconsider. Also, is the group really large? He might be overwhelmed by too many kids, noise, chaos, etc. At grandma's he doesn't have lots of other kids around competing for attention, making noise, bumping into him, etc.
It just may be that he's too young still. You may need to drop this and work on his separation anxiety by having a non-family babysitter come to your own house for short periods, building up your time away. If you go to church or synagogue or whatever, that's a good way to leave him with a non-family caregiver for an hour at a time (in the nursery) while you are nearby but well out of his sight.