1. STPGA Jr golf, click on their website, go to tournaments, click on that, 3 options come up, Little Linksters, Junior Links and Junior tour. You can see the schedule there, summer there is a TON of tournaments. Back on main page click on membership, jouin ($99 annually). Then take the caddie test if you want to caddie for him. I recommend you caddie.
2. Beltway Tour, join $75 annually, look at the divisions and pick the right division and put him in. Their tournament are mostly summer Mon-Thursday.
3. Join First Tee, google First Tee Greater Houston, forgot the but it is cheap, they have session, fall, spring, summer, each session is about 8 classes and have to attend 6 of 8, once a member they have monthly tournaments all year, always at FM Law, great facility for kids, tournaments always on Saturday, fFirst Tee emphasizes them being in their own except for the very youngest.
With these 3 you can play multiple tournaments in a week if you really wanted. STPGA Jr is age based, Beltway Tour is skill based and First Tee is a combo.
Once you get experience their is US Kids, Texas Junior Golf, Legacy Junior Tour. Under Armour also has a tour for Juniors.