I just did a test booking from PHL to TPA and had no problems, got as far as the credit card information screen, I didn't enter a credit card for obvious reasons.
I'll assume you downloaded the latest version of Firefox and Chrome. Don't forget to download the latest adobe flash (
http://get.adobe.com/flashplayer/ DONT FORGET TO UNCLICK MCAFEE) and Java (
http://www.java.com/en/download/ )