One solution to this is to create a plain text resume to be used when submitting through an applicant tracking system. You can still perform some basic formatting in plain text (e.g., string of hyphens for a horizontal line, asterisk for bullet, carriage return at 60 characters to prevent erroneous line wrapping, etc.).

At the top of the plain text resume include a line stating something like "MS Word/PDF formatted resume available at http://ExternalWebsite..."