Kids don't have to watch from the fence in San Antonio: 2 local tracks have evidence of junior karts, family sessions or young drivers on track — not just arcade games in the lobby.
Height matters more than age at most tracks — junior karts typically need drivers around 40–48 inches tall, and requirements differ venue to venue. Check the track's page below for specifics, and confirm with the venue before you drive out.
K1 Speed San Antonio, Andretti Indoor Karting & Games San Antonio — all listed below with ratings and contact details.
Go Karting USA lists 2 operating tracks around San Antonio — see the full San Antonio guide for all of them, not just kids go-karting venues.