I'm doing a project on scientific balloons, and I can't find anything on legal issues that the missions may face. My project consists of a hypothetical mission launching over Antarctica if that helps anybody.
Yes, but it's not so much legal "issues" as it is a simple legal requirement of getting approval to maintain safety in the air. As @BigMoneySeth's link shows us, an unmanned, free balloon must be registered with the FAA. Ok, but that doesn't help us in Antarctica just yet, since Antarctica is a mishmash of nations operating under a treaty.
So we need to go to a very boring document: the Interagency Air Operations Manual for Antarctica. This has an entire section on Unmanned Aerial Systems (which balloons count as). Basically, you need to file your plans with the base from which you are launching/where you are based (which is almost certainly McMurdo Station) and you need to be in constant contact with the air traffic control so that incoming/outgoing manned flights can avoid you. It's fairly simple stuff (make sure we know where you are at all times so that people are safe). You also need to make sure that your balloon complies with FAA regulations, since you would be under the law of the US.