After going through all the standard troubleshooting like rebooting and making sure all versions were compatible, I mentioned the issue to my co-worker. Fortunately for me my co-worker had already experience and spent time figuring out the issue; when I heard what the problem was I was glad he was the one who did the leg work figuring it out. I didn't do that much work with Excel and Access so I wasn't sure when the problem had begun.
Turns out a relatively recent Microsoft Excel security update ( KB958437) was the culprit. As soon as I got that uninstalled copy and paste between the two apps started working again.