It's not directly possible, unless you can persuade the user to install an activeX control, because...
In order to dial the phone, you need access to a Modem. The modem dials the number, and then the user is told to pick up the phone, at which point, the user would hang up. Old programs which would do this (and I can't remember any off the top of my head) would use this sort of sequence.
1) Do you want to dial the number?
2) User clicks "Yes"
3) Open Serial port (to Modem), send characters AT (response OK), AT DT 0123456789012
4) Pick up the phone
5) After a specified delay, hang up the modem
In order to get access to the modem, you'd need either an activex control or a java app. I don't know enough about either to do this!
Alternatively, there are services (which I can't find with a simple google search right now - but I'm sure if you're creative you'll find them), where a third party server dials the user and the business and then connects the two lines together. The third way is to take that in-house, and make it so that the user enters their phone number into a form, that goes into a queue at the web site, which is then reeled off to the PBX at the webserver end.
Lastly, you could use a VoIP solution like Asterisk, and then establish SIP connections in between either the business and the user, or between two users, well, let's just say with VoIP, the world is your mollusc.