Julie,
Thanks for your great question!
The answers are NO, and NO! Good news, right?
Because
Chef Mickey's and
1900 Park Fare are both located in Resort Hotels and not inside of theme parks, you do not need to have a theme park ticket to dine at either Also, you do NOT have to be a Disney Resort hotel guest to dine at either of these restaurants. Anyone can go.
I do strongly recommend that you have an Advanced Dining Reservation to go to either. They are just too popular to get into without one. To get these reservations, all you need to do is go to the respective restaurant web pages (linked to above) to make those plans.
I would also suggest that if time allows, enjoy exploring these resorts before or after your meal. They are wonderfully themed, and they are worth the time.
Have a Magical (and delicious) Vacation!