I agree with the suggestion about shipping your skis and boots, especially if you are not going to be using them after the Vail portion of your trip. Are you staying in LA for awhile or just making a connection there? If you are staying at a hotel in LA perhaps you could talk to them about having your equipment shipped there and stored as most hotels do have some sort of luggage storage facility. Hikers on the Appalachian Trail often ship supplies to US post offices that are in towns easily accessible to the trail so they don't have to carry everything with them. A lot will depend on your actual itinerary and whether you are stopping over in LA or just making a connection. You'll likely need to do some research and maybe think outside the box. I know it's always much nicer to ski on equipment you are used to but renting might end up being your best bet and the least hassle.
The storage lockers were a “pilot,” the kind of small test that city government frequently uses to test a new or controversial idea. The city offered up the lockers for individuals to use for months-long stretches. At the time, city officials warned that “misuse of the lockers, vandalism, or other unanticipated results,” could force them to cancel the project.