Hi Andrew,
Thanks for reaching out–we’re thrilled to hear that you’re considering The Events Calendar for this client project! Unfortunately, I’m afraid that none of our plugins currently offer this type of poll-based scheduling functionality right out of the box. 🙁 Adding on this feature to your calendar setup may be possible via custom development, but the level of customization required would likely be fairly extensive. As always, our Themer’s Guide and Technical Docs are available to users who are interested in customizing our products, and we’re happy to provide what limited guidance we can in our support forums! Please don’t hesitate to let us know if we can help in any way.
I’m so sorry to be the bearer of bad news here! We truly appreciate your interest, and we wish you the best of luck on your project! Thanks again, and let us know if you should have any other questions!