As I understand it you do not have direct access to the Internet but you probably have a connection via a proxy. That proxy is forbidding access until you authenticate yourself.
So in ASCII art:
PC1 ---- [-]
[s]
PC2 ---- [w]
[i]
PC3 --- [t] ---------------- Proxy ------------ Internet
[c]
PC4 ---- [h]
[-]
When you try to reach the Internet the proxies probably hijacks your connection and redirects you to a landing page where you can enter a username and a password. After that it remembers your IP and allows that IP access to the Internet.
Alternatively it could use some other authentication.
If this is correct and it allows one IP then adding a wireless access point complicates things. It will see the IP from the WAP and everyone on the WAP will get access.
Same diagram, with a W.A.P. added.
PC1 ---- [-]
[s]
PC2 ---- [w]
[i]
PC3 --- [t] ---------------- Proxy ------------ Internet
[c]
PC4 ---- [h]
[]
[-]
WAP ---- [-]
PCx now uses wireless. The proxy intercepts. User X enters username/password and the 'WAP PC' gets access. And via it everyone on Wireless!
I am going to assume that this is undesired.
You could solve that by using 4 access points and each of you only having the WPA2 password for your own access point.
now if the setup uses more complex authentication then you are in luck. It might not allow you access unless you have a cookie set. Or maybe it builds a tunnel, or..... regardless. If the access barrier is smart enough to only recognize your PC then all you need is that single W.A.P.