The short answer is: Of course!
If you're on the free plan, you can use your bounce.house domain name for your website forever.
However, if it makes sense for your business to have a custom website domain name - you can upgrade your account to easily register a custom website domain name through Bounce House, by following these steps.
Learn more about Upgrading your account.
If you have already purchased a domain from a third-party provider and want to keep it registered with them, you can connect it to your Bounce House site by following a process called domain mapping. Follow these steps here.