What exactly do you mean by “appropriate playgroup?” We have a group on Thursdays which is all about learning more about different games, which you and your son are more than welcome to attend. Please note, however, that we are currently all adults, and we play in the back of a bar. Friendly adults and respectable bar, but still.
If your son does wish to drop by, please let us know in advance in the weekly thread in this forum. As bonkers mentioned, we do sometimes deal with themes that I don’t feel would be appropriate at that age. For example, not long ago we had a game about racism which I feel would be a bit much for most 13-year-olds to handle. By giving us a heads-up in advance, we can make sure the subject matter is age-appropriate, and also plan on not playing until midnight.
(As a side note, I do not believe a minor needs the owner’s approval, at least until 11 pm.)
Otherwise, just playing with friends is always the best way to go. Perhaps pin a note on a bulletin board at school?
And if you’re interested in conventions, the Don’t Panic Con is quite family friendly. The main focus is board games, but there’s also plenty of role-playing. The next one should be in February.