The simple answer is maybe.
If you are starting from scratch and providing all documents to the lineage society, the same supporting documents generally can be used to join another lineage society. Often you’ll need to add a few documents. For example, the Daughters of the American Revolution currently has a (slightly unwritten) rule that if the death certificate contains the date and place of birth and the names of the parents, the applicant doesn’t need to submit a birth certificate. But if you want to join one of the Colonial Dames societies on the same paperwork, that birth certificate will be required.
But if you are relying on previously submitted documents, there are only a few societies that will take another society’s paperwork. The best known of these societies is the Sons of the American Revolution (SAR). SAR will accept DAR record copies as supporting documentation with a few caveats: simply put, they must be post-1985, and they should refer to original documentation rather than to a previous member’s application, so that the verifying genealogist can see the sources used. See https://members.sar.org/media/uploads/pages/77/Y4SVnQ7JoiU4.pdf for details. A few smaller societies will also accept record copies.
So, in short, if you submitted an entirely new application post-1985 to DAR and would like your son (or male family member) to join SAR, go ahead and submit your record copy. But if you didn’t start from scratch, proceed with caution. Work with someone who truly understands the process.