My wife is in supply chain mgmt, and as others have said, it's a wide ranging discipline that's applicable to many facets of the ski industry. Vendor relations in particular seems like a good place to start - obviously from a manufacturing perspective, but also at resorts for items like food service, payment processing, providers/operators of the tech for RFID pass management, and even just ordering pens for the office...the list goes on and on, and will be different for every resort/company. It's really a ton of work.
For internships, if you're willing to go unpaid (and you might have to, as I'm sure a lot of people want in the ski industry, and it's pretty small, relatively speaking), try stalking some people with Operations/Supply Chain/Logistics related titles on LinkedIn and just offer your services gratis. They might take you up on it. A lot of SCM teams are pretty bare bones. As you said, it's often overlooked, both from a budgeting and staffing perspective, so free or cheap help would likely be appreciated if you can find the right people to ask.
Intrawest, Vail Resorts and Spyder are all headquartered in the Denver area and have considerable administrative presences. Are you willing to move for the internship?
Edited by LiveJazz - 11/9/15 at 3:33pm