OP, I don't want to downplay any MH issues. But I've noticed a lot of children - especially the more introspective, less sporty ones, go through a sort of cocoon year where they barely leave the house. Both my DC did and so did I. Usually around age 12.
If I were you, I wouldn't worry. Make sure there is a variety of activities at home that he can do.
With my two in their chrysalis phase, I just had loads of conversations with them about fantasy futures. Like - what would you spend £1million on? If you had to spend it all on yourself? If you had to spend it all on other people? What would be your dream home? Dream day in your life? Coolest outfit or hairstyle the'yd love to have. I was surprised how enthusiastic they were to discuss this stuff. And I got to know what mattered to them. It made it easier to gently develop the conversation into suggestions for skills to develop or things to research..
The summer they are 11/12/13, they are in the process of transitioning from child to teen and they really don't know how to emerge. I used to tell DC they had to do one thing to keep them fit each week and one thing that helped them build a skill they wanted. At least one of those two things had to get them out of the house into the world. But they could choose which. And the rest of the time, I let them mooch.
I also helped them think about ways to earn money over the summer and what they wanted to save up for. Your DS could offer to look after neighbours' small pets when they are away on holiday for a few pounds a day, for example.
Allow him a bit of time just to be. At this age it really is helpful. But keep an eye out for ways to help him emerge from his shell. They do hate parental advice, so I used to hunt around on Youtube for cool young men who were handing out the same advice as I was trying to, then show them the video. It often worked. A coool young man doing workouts, or explaining how to lift weights, or how to handle a tricky bass guitar line, or draw a manga character etc.
I told DS2 he just had to learn to swim, non-negotiable as it was a life-saving skill. Despite endless lessons at school and summer programmes he still couldn't, aged 12. So I dragged him to the pool once a week but just made him swim for 10 mins, mess about for 15 then out the first time. We stepped it up every week, and then he saw a local challenge where you could clock how many lengths you swam. Within six months, he could swim a mile. Something like that, where he can see an improvement - could be weights, or running or cycling, or learning guitar or writing a comic book or fantasy novel.