Yes, and you've answered the question in your question! As you note, the FAA regulates launch and reentry for US-based operations. As part of this regulation, the FAA sends around applications to the U.S. Government interagency for comment before approval. One of the questions they ask is whether there are any planetary protection issues with the proposal. Here, the issue would be "backwards" planetary protection, ie theoretical contamination of Earth's biosphere. I doubt any US government agency would object to a SpaceX sample return, provided that they weren't doing anything overtly dangerous, but there would at least be permission required, and it would be done through the reentry authorization handled by the FAA.
Details on backwards planetary protection standards: COSPAR is an international organization of scientists that, among other things, develops technical standards to prevent both forward and backward contamination. See one set of standards here. Members of NASA's SMD participate in COSPAR, and NASA bases their own internal protection standards on those recommended by COSPAR, with some modifications. So here's how planetary protection concerns are addressed, either forward or backward, in detail: the entity (here, SpaceX) seeking the license requests an authorization from the FAA. As part of that application, they include info on planetary protection issues that might be relevant. FAA circulates that application to relevant US Government agencies, who then comment. NASA planetary protection experts look at the data and provide their input as to whether the measures proposed follow, or are equivalently protective, as those NASA itself would follow (which are based on COSPAR recs as I say above). The Department of State says whether or not these measures meet the baseline legal requirements set out in the Outer Space Treaty. FAA then decides whether or not to issue the license, or they could require changes, etc.
As part of the interagency review, the U.S. Government (mostly the State Department for these questions) would also examine whether there are any other foreign policy or international law concerns. As some comments have noted, the U.S. position is that space resource utilization is permissible under the Outer Space Treaty, so there shouldn't be concerns with sample return in that sense. Planetary protection concerns are part of this inquiry, as described above already.
(PS other authorizations, such as for communications via the FCC, and remote sensing via NOAA, would also look at similar issues, but here those inquiries would likely just focus on the FAA reentry licensing.)