Actually, your suggestion about focusing on posting interface plugins helped. This is the plugin that turned out to be the problem:
GTQuickreply Smilies Popup GTQuickreply Smilies Popup - vBulletin.org Forum
The weird part is that it has been installed in my forums for a long time without any conflict. I guess some change in 3.7 at some point caused it to interfere. Thanks for your help.