Any suggestions for the software used to make this video?
The drawings can be made in illustrator, but I can't figure out how to animate it.
It honestly looks hand animated to me, however the digital equivalent would be to start with vector graphics and animate them in something like Flash or After Effects. Either product is capable of keyframe animation of various assets including vector graphics.
Personally, I'd rank them by likelihood as by hand and scanned, followed by Flash, followed by After Effects as the possibilities. Keyframe animation allows for objects to be set at a start point and an end point and the computer fills in the values in-between.