Why not use PMs? If you want to make it easier for your community, you could share with them a pre filled url with all the info: How to create a new personal message via a link with pre-filled information
For example:
https://yoursite.discourse.org/new-message?username=admin&title=Contest%20Entry&body=Please%20add%20your%20entry%20here.%0A%0DName:%0AOther%20details:
And the link would open like this:
Maybe polls would work: How to Create Polls
Or you could integrate Discourse with a survey third party service like SurveyMonkey.