Are you looking for a package which will do this for you, or for help with programming a solution yourself?
If the former, I would suggest Google, using search parameters of
+DOS +"Serial Data Acquisition" or, contact the weigh scale manufacturer. One package which might work for you is
TALtech's File-Wedge.
In either case, if there's
ANY possibility of getting a Windows machine into the picture, it would be very beneficial, as there are a LOT of products available for windows programming, and data collection. If nothing else, you could add a windows computer with 2 serial ports to read the serial comm from the weigh scale and echo it to the existing computer. That way, you could sample it and store it in the data file. (assuming that you have networking enabled on the DOS machine)
If you cannot get the DOS computer changed over to windows, or adding a monitor per above, I would suggest taking a look at some of the programming archives - Turbo Pascal, for instance, had a lot of support in the 80's/early 90's, and I remember quite a few programming examples dealing with serial I/O. You'll be on your own as far as the dbf file and index updates, though.