Welcome, Kristin!
I hope you are having fun planning your vacation! Thank you so much for letting us help by sending in your question.
The Walt Disney World® Resort Theme Park closest to
Chef Mickey's at Disney's Contemporary Resort is The Magic Kingdom. From there, you can get to your dining reservation on foot via the walking path from Magic Kingdom to Disney's Contemporary Resort. You will then take the escalator or elevator in the lobby to the 4th floor Grand Canyon Concourse where the restaurant is located.
The second option is the Resort Loop Monorail. The Monorail Station is located near the front entrance of Magic Kingdom and it's first stop is Disney's Contemporary Resort. You will be able to see Chef Mickey's as you exit the Monorail Platform. If you utilize the Monorail option, just remember you need the Resort Monorail, not the Express Monorail. In my experience, walking over is usually faster.
I hope this is helpful, Kristin! Please let us know if you have any additional questions. We are here to help you navigate your way around Walt Disney World® Resort.
Faith