Do those URLs actually resolve to the same site/server? IMO, the easiest way to do this is to do it as part of your build process. When you publish the app for "client A", you use the web.config (etc) for that client - and ditto "client B". May SCM systems allow this type of per-client configuration, or it is pretty easy to do it with a few build scripts.
The alternative (if they absolutely must be the same physical code files on the same server) is to move this into the machine's web.config - much more complicated, but you can use the "location" node in web.config to specify connection-strings per IIS site, even if they resolve to the same files. I stress that this is risky, hard to verify, and easy to get wrong - the build process is my preferred option!