Hello. I require all of my students to wear boots and long pants when riding (make sure your child knows that...shorts are absolutely forbidden, unless he/she wnats to tear up their legs and get pinched constantly). I also forbid sneakers/tennis shoes/gym shoes. I generally disapprove of SHOES in general (meaning anything that does not lace/zip up over the ankle), and always try to steer my students away from buying anything other than paddock boots or tall boots. Fashion shoes are not meant to be ridden in.
Also, work boots or any other RUBBER or RIDGED sole shoes are dangerous to wear while riding. The treads on the bottm will "grip" the pad of the stirrup such that the rider will be unable to to pull their foot free from the stirrup in the event of a fall (or "unscheduled dismount" as we call them here).
A proper riding boot will have a hard sole (not rubber), and a heel NO GREATER than an inch or inch and a half. I simply want to scream when kids show up to summer camp and are wearing those high heeled fashion boots that vaguely resemble a tall riding boot...or sometimes they wear WINTER boots...ugh.
Gloves are ALWAYS appropriate, whether showing or not. You can buy a nice neoprene palmed or leather palmed glove from a tack store (or the closeouts going on now at Stateline). They will help you grip those sweaty reins. They don't have to be black leather show gloves. I have a nice leather palmed/crocheted bakc pair I wear when riding.
Also, instruct your child to wear tall socks (should reach the knee) under his/her pants. This will help alleviate pinching from the stirrup leathers.
If the summer camp doesn't provide helmets, you can buy one from any Tack shop or online/catalog outlet for $29+. Helmets are not negotiable at my barn. No helmet, no ride.
OH KAY: Where is your kid going to camp? I am originally from LI and would love to know. I can recommend many nice tack shops on the island that can outfit your kid.