Hi Jason,
No, the WebStore is a very basic ecommerce feature that currently supports download products fairly well but not much else.
You might take a look at my add-on product Event Calendar Pro, which can sell tickets for events. It supports a free form text area and an instruction block where you can ask users to provide additional information. You can try it out on our demo site to see if it meets your needs well enough.
Hope that helps,
Joe