You do not need any special gear for rugby, just a pair of cleats and a mouth-guard.
For cleats you can use regular soccer cleats. The issue with most football cleats is that they have a toe cleat, which is not allowed in rugby games. When your player gets older, it may be worthwhile to look for a “rugby boot”, especially if he/she is playing in the “prop” or “lock” position. In these positions, traction is essential and if you play on soft ground then cleats with bigger metal studs can be helpful. Having said that, soccer cleats are always an acceptable option.
For mouth-guards there are many options and you can find them in any sporting goods store. There are no particular requirements for rugby except that it is important that the player can talk while wearing the mouth guard. This is important so that he/she can communicate with his team mates during game play. As such, many mouth guards designed for football are not good for rugby, since you cannot talk while wearing them. Here are three common brands that you can find on Amazon: Shock Doctor, OPRO and SISU. They normally come in a junior and an adult size. For children with braces, there are special versions of Shock Doctor and OPRO, while SISU offers special instructions when molding on top of braces.