It's possible, but not the way you think.

The first is the top level domain. There are a couple of options. One is you'll need to DNS redirect from their web hosts to your address. Note this is not an HTTP redirect! If your servers take over the DBS hosting process, then you take control of that, which is something some CDNs do (such as CloudFlare). However, given you're asking the question, I'd suggest it might be too advanced for you if you've not got a good grounding in web networks.

The URL rewriting element will give you the rest of the path though. Indeed, if you're using MVC you get this out of the box through its routes (you just direct their path to your path). But the URL rewriting module is available for IIS in any case.

If you need any more help or info, happy to schedule a call in. Just ping me a message.