if the old DOS application does everything they want, maybe as an alternative to rewriting it in Access or whatever, try running it under DOSBox (www.dosbox.com) or some other DOS emulator and see if that works for you.
I think DropBox is mainly used these days by people wanting to play old classic DOS games, but perhaps it would work for you.
also see: Export all tables data to CSV, JSON, XML, etc - #15 by FedericoNavarro