You are welcome to dine at any of the restaurants in the resorts or at Downtown Disney without using a park admission ticket. This includes Chef Mickey's at the Contemporary.
Guests who have used a travel agent to book their hotel accommodations are certainly welcome to use the online dining reservation system. See
HERE for information on how to do that. However, because you booked through a travel agent, you are not currently able to use the 90 + 10 perk when making online dining reservations. Read more about this
HERE. If you are making dining reservations fewer than 90 days in advance, you can go ahead and do this online, you just won't be able to link the dining reservations to your room. If you prefer to make dining reservations over the phone, you can do this at 407-WDW-DINE.
Have a great time at Walt Disney World.