This problem has appeared and gotten worse rather quickly. When I picked the car up, it ran and seemed to shift smoothly (as a 914 can shift). But within a couple days, it started to become difficult to get it into 4th. Now it's difficult to get it into 5th as well. If the car is standing still, it seems like it will go into all the gears as it should. So before I take her to the shop, any advice on anything I can do? Do you think this is a clutch issue? Shift linkage?

Had the same problem and replaced all the shift bushing and it sounds like the shift coupler bushing is the culprit. I would recommend replacing all your shift bushings.
http://www.pelicanparts.com/catalog/SuperC..._pg3.htm#item10

Good Luck

Besides the obvious bushings at the cuplers which can wear and disintegrate i'd first check the cone set screw at the coupler right after the firewall.
When that one comes lose you'll experience exactly what you describe.

Note that those screws are one time use only, they have a built in metal crush locker.

Jwest sells a needle screw with a jam nut. They are reuseable. Buy 2.

Failing bushings die fast.

Thanks everyone for the help!