Letterhead is a stupid requirement now that things are digital, because it's
incredibly easy to whip up some digital letterhead. Just put your school's logo, maybe a line, and your prof's department in the header in Word, and boom. letterhead. Just has to look fancy and show their affiliation.
Most people have methods of signing electronic documents now. There are entire apps dedicated to easily signing .pdfs, but for letters it's even simpler: they're drafting them in Word. Have the prof take a photo of their signature, paste photo into Word at the end, save as .pdf with the letterhead and the signature already included.
Or, if they want to be old-school about it, draft it, print it on official department letterhead, sign it, and scan in the whole thing to upload.

To each their own.