I moved my microSD card from my old moto droid to the droid 4, but I can't see any of my old SD card apps. Is it possible to import them? Neither phone has even been rooted.
By simply moving the SD card between devices, it is possible to transport data. The apps you are referring to are probably the once you moved to SD card via App2SD -- this will not move the entire app, but (hopefully) the biggest part of it. At least a small piece remains on the device -- which is, amongst others, how the device knows about those apps. And which is why "the other phone" won't show them: It simply doesn't know they are there, as they are not "registered" with that device.
So if you want to move them over, you will have to re-install them on your new phone (here: the Droid 4), and then (optionally) delete them on the old. The apps data are lost that way -- unless you can export them from within the app on the "old droid", put them on SD card, and import them with that app on the Droid 4. Or, alternatively, if both Droids are rooted, and you use Titanium Backup to backup app and data on the old, and restore them to the new phone.
If your idea is to permanently switch the card to the new device (i.e. not just temporarily, and then switch it back to the old), you might want to follow these steps:
Alternatively, if your old phone has sufficient internal memory, instead of uninstalling the apps (step 2) you could simply move them to internal memory. Then, if you configured your phone to backup apps+data to the Google cloud, you should be able to just "sync them back" to your new phone (in step 5). I've never tried that, but the possibility exists.