With a budget of under $2000USD, would it be possible for an undergrad student to build a muon scatter detector (drift tubes or another method)?
This paper on arXiv seems to be exactly what you're looking for, so the answer appears to be a resounding yes.
The design uses an inexpensive plastic scintillator and a silicon photomultiplier for light collection, and the total cost per detector is claimed to be around $100 which, as far as I can tell, is very plausible.
That being said, the detector itself is not the only thing you'd need. At the very least, you'd need the electronics necessary to read the detector output and some software (which you could probably write by yourself) to interpret those readings. You'd also need to calibrate the thing, which might take additional resources unless you have ready access to well-characterized radioactive sources. This would take a bit of extra money, but I can't imagine it'd be even close to your stated budget of $2000.