Maybe update FF to the latest XP-supported version, 52.9.0 ESR? I have that, along with Chrome 49.0.2623.112 & Opera 36.0.2130.80 in my XP test partition and they all work fine.
Out of curiosity, did you add recently or already had a Script blocker add-on? If so, could it be it's the one misbehaving and causing this?
Also, here's a few temporary workarounds you could do, so at least you can get to the pending messages:
1. Go to / create a bookmark to https://groups.io/groups , or click on the "Groups.io" link on the upper left of the screen, and in there, any pending messages will show under the ApprovalsNeeded column as a link and you can go to them this way.
2. Create a bookmark to https://groups.io/g/yourgroupname/pending . This will take you there and should also expand the Admin menu on the left so you can get to the other admin options.
3. Create a bookmark group in the toolbar and add in there all the bookmarks that take you to the groups and/or specific group areas/functions, sort of replicating the "Your Groups" menu. (you may want to do that anyway, I've done it because it's faster than getting to something the "normal/regular" way)