Have the kid do sports because they want to do sports. If they are in an introverted you may never be able to bring them out of the shell anyway.
But you do have some options to help with your fears.
First selection: Team or Single person sport.
Second selection: League.
If your kid is more of the loner type they may prefer something along that lines. You still are forced to interact with people but there is no peer fear for failure.
I was not really a loner but I did not like the thought or people relying on me or the other way around. I preferred swim club, martial arts, and BMX to team sports.
There are also a great many options in terms of league play and skill.
Rec park offer great “get involved but not-so-competitive” leagues… While on the other hand SDR has a kid playing at the almost “Semi-Pro” level league.
So you should be able to find something that fits your kids personality and comfort…