Tatin - had a similar issue a few years ago. You may already have tried these, but what appeared to solve the problem for me was a) make sure your Pop-up blocker is off. If it is already off, turn it on and then off again b) make sure you don't have any active Ad-blocking software c) if you have Anti virus software, put the TRIPS URL in the 'safe site' list d) just prior to accessing the Staff Travel site, clear all cookies from the cache on your device.
I'm not sure which of these fixed the issue for me, but one or more of them certainly did and I've had no problem since. Sorry this response is late - only just saw your post.
Good luck.