If it needs to be compiled and you don't have the necessary access, then your hosting provider might compile it for you. If the extension is a popular one (from PECL perhaps) then chances are there is a version available pre-compiled that you can just throw into the extensions folder (its location depends on the setting in your php.ini), enable it (again in php.ini) and restart the web server to load it in. You're probably out of luck if it's your own extension and your host won't give you shell access (note: you don't necessarily need root access).
Your use of the word "custom" in this instance seems to be confusing matters. Custom as in home-brewed, or custom as in not-installed-yet?