Yes this is possible. Use a very good primer; either an oil base primer, like Zinsser Cover Stain, or acrylic, like Zinsser 1-2-3, tinted light-medium tan. Make sure to thin the primer a little as to not have too much build, this could fill some or all of the embossed (wood grain) areas making staining very difficult.
Another option is to fully strip the door and apply some fiberglass resin to the "damaged" areas. This is a bit tricky but also doable. The resin is clear so priming might not be needed as long as the original color is intact.