I was a shy kid (60 years ago!) who didn't ask anyone from school to come over, except once, when I got up the nerve to ask a boy I had a crush on to come over, on our daily walk home from school. "To do what?" he asked. "Oh, nothing much - just play, I guess," I said, taken by surprise without a plan. "Well, I guess not," he answered, and went to his own house as usual.
Actually, I was a lot happier to relax with my brother and his friends, than I would have been with the challenge of entertaining anybody I would have to face in school the next day.
My point in writing this is to say that in spite of this unpromising start (which worried my mother, just as it concerns you), I found, when I went to a very small high school, that I could get to know my classmates well enough to hang out at their houses and have them come over to mine. I'm sure this was partly my own maturity finally kicking in; but the intimacy of the little school, where every student was known and valued for his particular contributions, was very important.
When I went away to college, I was able to draw upon that experience and take advantage of all the opportunities to be outgoing - performing arts, even! - and make as many friends as I could possibly keep up with.
So my advice is first, Don't worry too much about it at this stage, and second, Consider an occasional organized activity with your son, let's say, a trip to a science museum or a show of some kind, and suggest that he invite one friend who would be particularly interested in doing that. This would give the event the safety of your being there, and the structure of an activity where the focus wouldn't be directly on him.
Good luck! R. L.