Do I have to own a company to rent equipment?

You don't have to own a company to be a renter on Gearbooker. You can rent as a private person or on behalf of a company.
However, you actually always run through the verification process as a human person, hence the requirement to provide your ID and some other personal data. Of course it is possible to rent equipment on behalf of a company if you need an invoice for your company. If you enter the VAT and company data in your account, you will be able to download a booking invoice on your company name.