My son definitely has the Aspergers low muscle tone and social awkwardness. He's had occupational therapy for low muscle tone and fine motor issues, and isn't good at (and dislikes) sports. He has social awkwardness, but with ADHD isn't shy or quiet at all. So I don't think shy/quiet is necessarily the same thing as socially awkward.
Sometimes it's hard being a quiet mom of a loud kid. I've always been a shy bookworm, not good at sports. I've wondered over the years if I could be on the spectrum myself, or if it could have been because of my nearsightedness (hard to play sports when your glasses don't fit well, hard to make friends when you can't see their faces) or allergies(hard to learn how to swim when you're having trouble just breathing).
Perhaps you could find some forms of exercise that fit your personalities better than team sports. Gardening, nature walk, hiking, swimming, biking, riding a scooter, weightlifting or raquetball(indoors and not too many people).