Qualifications
Knowing the greeting card industry is the first qualification. You know those cramped stacks of cards in the local supermarket? Meet your new after-school hangout place.
Knowing the different conventions in a "Sympathy," "Love and Friendship," "Good-bye," "Thank You" and "Congratulations" card (and whatever else you discover in the card aisle) is key to knowing where your own work fits and which publishers are most likely to accept your work. Don't submit your most beautiful, Shakespearean poetry to a greeting card company that specializes in fart jokes.
There are around 2,000 greeting card publishers in the United States (source). Many of them will post guidelines for submissions and detailed information on how to submit. For reference, see this one. Be smart about where you submit your material, and above all, follow the directions. (This paragraph is the first test.)