Rachel,
You absolutely can make a reservation to dine at one of the restaurants in a Disney resort, even if your "home away from home" is not a Disney hotel! My family loves
Chef Mickey's, and it is one of the most vivid memories from our first trip as a family for my youngest daughter.
There is a situation where the system will allow guest of Disney resort hotels to book a bit further out, as a perk to staying at a
Disney resorts collection hotel. They are allowed to book out 10 extra days, in addition to the 60-day window that exists for all guests. Could it be that you are looking outside that window?
Based on your arrival date, you should be able to book dates now through July 1st or 2nd of your trip. My recommendation is to try again and write me back with more particulars of the date you want to dine, and what kind of roadblocks you are running up against. If the problem persists, I am sure that a member of the Disney technical team will be able to restart the magic!
Write me back to let me know how it goes?
Have a Magical (and delicious) Vacation!Doug