I am not a parent so it is hard for me to relate to your concerns; however, I am a daughter

If I may suggest, perhaps you should speak with a T to talk about why it is so vital to you that your daughter have a big social life, when it seems she is doing fine. I had some social anxiety as a teen, and my mother pushing me or even forcing me to do things (because she thought it would be good for me) only made things worse.
One of the things that stood out for me is that your family moves quite a bit. Perhaps that is the reason why she doesn't feel comfortable getting close to non-family members and she chooses the safety of her brother and being at home (she can count on you to be there and you "know" each other). Just throwing it out there, though. Idk. I am not in her shoes.
I do remember very clearly that teenage years are soo hard and if your daughter is naturally introverted or simply comfortable being in her own skin, in her own company, that isnt necessarily a bad thing. Esp because you mentioned she does well in school, goes to school, is on the swim team and is generally a good kid.
Just a thought, too, from my own pre-teen, teen years: a lot of the girls were incredibly gossipy, catty, hormonal and some times cruel. I saw right through them and didnt want to be around that in social situations. In addition, I did not want to particiapte in the "experimentation" the kids were involved with. this type of behaviour crosses all socio-economic areas, and I was raised in a very well to do, upscale, family-oriented beach community (in case you were, wondering).
I also attended summer camp, sleepaway, for two weeks every summer, rode horses as a regulae hobby and loved water sports (from the time I was little). I participated in activites during the school year, as well. And we went in family vacations.
I just didnt want to get involved with all the teen drama and antics. And I was a teen
I wish my mother, though, understood better instead of nagging me and pushing me. I felt as if everything was ok ... she was the one that made me feel abnormal and not good enough. It cut me deeply and affected my self esteem.
Please let us know how you are doing. I really do feel for you.