To be technically correct, think about this:

Ходовая часть автомобиля включает в себя раму, подвеску, задние и передние мосты, колеса и шины. Входит в состав шасси.

Now i can translate this for you.

Ходовая часть (ходовая is a shortened term) consists of frame, suspension, back and front axles, and wheels. It's a part of chassis.

As far as I understand, it can be with or without frame depending on vehicle type (some cars don't have frames, the parts are fitted to the body in them). But it should include suspensions and wheels.

I don't know the exact English term for it. Maybe 'running gear' , or also 'chassis'. Literally ходовая часть can translate as 'running part of vehicle'.