The driveway is relatively new (2-3 years old). The installer did not provide a new foundation and the driveway is now sinking in places. How might I fix the gaps at edges of my home's blacktop driveway?
They may have done poor preparation and installation work, but the gaps here look like a result of the brick moving rather than any kind of failure or movement in the asphalt.
There exist crack seal caulking products that you could use in combination with foam backer rod to fill the gap. But if the brick edge has moved, it's likely to continue moving, and soon enough the crack seal would fail too. (To find these sealants, do a search for "elastomeric caulk for asphalt," "self-leveling asphalt sealant," or similar.)
Personally, I wouldn't do anything. Whatever you do will look like a patch job, and it won't stop the rest from deteriorating. Not worth the effort.
If you're determined to do something, consider removing and reinstalling the brick border properly. That will close up the gap and the brick will itself look better.