Leave empty field in the email to fill it with a special code, people have paste that code here and a random winner will be chosen after getting 100 responses.
you can pay with bitcoin.
Obtaining email addresses wouldn't be really a huge deal especially if you have dummy addresses and would not be clicking any links you might receive. Some researchers often use email addresses as an evidence and somehow a proof of legitimacy (of his/her findings). It would be less worse than requiring Personal Info right? But yes, a simple screenshot of response is enough. Yet if it would be effective to have email as a proof of those responses for future references, then there isn't a problem at all.
Will respond on the form, yet it would be better once there is an escrow for the prize. Also, @OP, you can send me a PM if you really need help responses. I have tons of connections to many groups. I guess that could help.
EDIT: A little suggestion: AFAIK you can limit responses based by user's IP or what. I guess having an option to create multiple responses to the form might ruin your findings as for the fact that "one paid person can make one thousand responses, enough for being used as data, yet isn't credible.".