MacBook Air not contacting Wi-Fi with NETGEAR Extender

If the login functions through a Webpage, it's probable that the closest you could come is to write a script to fill in the form for you.

If you're lucky, the login form might use a HTML GET request so you could to bookmark the page that logs you in. Pay attention to what redirects occur once you login. If you see something like this:
http://example.com?name=fred&pword=qwerty, you can bookmark that and go to it to log in automatically.
This is unlikely, though, because it isn't a very secure choice for your company. The form probably uses a POST request.

If you have checked both the user keychain and the system keychain for old passwords, we're left with obscure guesses.

Two things come immediately to mind (from the obscure side).

1. Your mac will join any base station with that SSID. I have seen people literally setting up a second router in busy coffee shops - you have to hold the option key and make note of the mac address you have connected to and see it change to catch this.
2. If your wifi connects to the router wifi, it doesn't mean the dhcp server has any more addresses to lease. You will need the owner to reset or power cycle the router to see if the leases are all out and unexpired. You basicall have a great pipe connecting you to the main, but no water is flowing to you until you get a lease.