The chassis is probably the second easiest part of designing and building a self-balancing scooter. There's a huge variety of ways to build a good chassis, so I won't go into too much detail, but I will cover some of the highlights.
Firstly, what the chassis needs to be able to do...
- Support the weight of the rider, batteries, and motors. Preferably without flexing or straining too much to do so.
- Keep the motors in a fixed position. The motors will naturally want to turn themselves opposite the direction of the wheels (Newton's first law: for every action there is an equal and opposite reaction). The less the motors are allowed to move, the better the scooter will be able to balance, and the rougher terrain it will be able to handle.
- Support the steering handle as rigidly as possible. The more securely mounted the steering post is mounted, the more stable the scooter will feel. It may not seem like it would make a big difference, and to an experienced rider it doesn't really, but to a beginner having something solid to hold onto is a huge bonus.
The largest factor in making all of this happen is material selection. I have seen (and sometimes built) working self-balancing scooters made from just wood, wood and metal, steel and aluminum plate, acrylic sheet and aluminum brackets.... The possibilities are numerous, and vary hugely in cost. A lot of it comes down to what you have available, how much money you're willing to spend, and what kind of materials your tools can work with.
Personally, I have done three scooters: one with a purely wooden chassis, one with a wooden chassis and metal motor and battery mounts, and my current version uses entirely metal parts. The all metal one is definitely the most structurally sound, but it's also the heaviest, most expensive, and hardest to work with.