Redirect but keep the original url in the address bar

The first approach is likely what you need; that is, replace the Host header and request URI path before it is submitted to the server. If you're running 11.5+:

when HTTP_REQUEST {
    HTTP::host "vendorsite.com"
    HTTP::path "/UserAccount/Login"
}

Is this not working as you expect?

Incidentally, you can also use a Local Traffic Policy:

• SOL15085: Overview of the Local Traffic Policies feature