Don't overthink it. Don't setup a company right now, you're not one - yet. If you want to setup an entity quickly with minimum effort then a sole trader would probably be the quickest way to make things 'legitimate'.
Yes make a contract, but there are dozens of sites to get a lawyer-generated contract from. Here's an example for my country: https://lawpath.com.au/
When I got my first few customers my "order form" was just a survey form (I think it was Typeform) to get the companies details, with a Stripe payment embed in it.
There's a lot of good info on here: