Are you using a serverquery login? This isn’t getting really clear from your question. If not, the site will most likely login as a sererquery guest and therefore will probably not have either subscription power or client view power (or both) and can’t see the clients. The list of channels and total connected clients is not covered by these permissions since these fields are contained, when the virtualserver is queried.
You could either go ahead and provide the website with the serveradmin credentials (printed in the teamspeak server log on first startup) but i DO NOT recommend this at all, since these credentials provide full access to every aspect of the teamspeak server. The other solution would probably be granting the according privileges to the server query guest. I have not tested the second solution but can imagine this might help.