This isn't possible via matrix.to, they only allow you to pre-set a small list of pre-approved web clients. I think you could maybe self-host matrix.to, to get what you want?
I haven't self-hosted Cinny, but I do have an element instance. Using that, simply copying the url from the address bar, works. That takes them to page that allows them to sign up on my instance, and then join the room/space.
Does this work with Cinny?
Alternatively, you can give them the link to your web client, and the room/space address, separately. That will both allow them to sign up, and then join the room/space using the address.