Hi chris69. Welcome to SitePoint.
You have a lot of options there. You could build the test site on your own server, or on your own computer, etc. and then move it to the client's server when ready. In that case, you don't need to ask anything of anyone until the site is ready to move. You could perhaps also build the new site right on the client's server itself. If a client has a static site, I sometimes just build the site right where it will exist but in the background—assuming it doesn't conflict in any way with the existing site.