Okay this might be a dumb question, but... does your company block those sites? Also, is your phone authorized to BE connected to the company wifi? Sometimes it can LOOK like you're connected, but while you're actually connected to the network, you're also blocked from the internet. Again, these might be dumb questions, since I'm assuming that the phone did work at one point, but I thought it couldn't hurt to ask.
Also, did you try and make it 'forget' that network, then log in and re-input the wifi key?
Phone calls and texts work because they're all going over your G3/4/lte connection. If you really need to get to the other sites -- and you have the download quota -- you could just turn off wifi and use the cell connection. Unless you're pulling down a bunch of pictures, it shoudln't be too costly in terms of bandwidth. OF course, if you only have like a 1g plan, it might be an issue.