I’m not sure a third party is required to hand over their stuff for discovery if they are not named in the suit................
The short answer is yes you can.
The long answer is that you have to makes sure the request is not unduly burdensome, which is probably why the DNC is framing their response the way they are.
Here's a good article on the subject:
https://blogs.orrick.com/iplandscape/2014/04/24/be-reasonable-with-non-party-discovery-requests-a-lack-of-diligence-can-cost-you/
So what are the lessons learned? Before filing a motion to compel document production against a subpoenaed non-party, make sure to request documents from the party before moving to compel, and more importantly, perform diligence to ensure you dont already have what youre asking for.
So better buy some popcorn and get ready for the hearing!