Yes, it's good for one year. I think it took a few weeks to get it in the mail, but according to All Ears, you could do this:
MEMBERSHIP AND CARD INFORMATION: Call Disney's Dining Experience office at (407) 566-5858, Monday through Friday between 9 a.m. and 5 p.m. Membership is not available on property. Membership is only available by mail or phone.
BY MAIL: Disney's Dining Experience, Post Office Box 10000, Lake Buena Vista, FL 32830
BY PHONE: (407) 566-5858.
NOTE: Allow several weeks for processing. If you will visit Walt Disney World before you receive your membership card, you can arrange to pick up a letter of authorization from a Guest Relations office. You will need to show a photo I.D. to pick up this letter.
That should ensure that you can use the letter on you first trip and you'll have the still-valid card for the second. HTH!