How should I dispose of the vinyl shed walls?
If you have municipal trash collection, you'll want to contact the company in charge of pickup.  It's quite common for them to have a website, which typically has information about how to handle various types of waste.  

If you're not sure who handles the collection. That information is usually available on the town/city website, or by calling the town/city.

If you take your waste to the dump (not sure if folks still have to do this), you could simply ask someone on your next trip.