Is there a way to create an anonymous form but still have the email entered for a chance to win something?

Yes, you may ask any information that you want and it's not needed to ask for personal information of the submitter, you may just ask for the email address and save it for your records so you can contact the submitter again. 

This link will help you to start creating your form: https://www.jotform.com/help/chapter-2-Create+Forms 

If you have more questions, let us know.