Apply for a job with Hallmark. Hallmark does not accept freelance, unsolicited or unagented material. In addition to filling out an application to work for Hallmark, you can include samples of your work and a resume that shows a track record of successful greeting card poetry with other companies. Working for the company as an employee is the only guaranteed way to write poetry for them. You can apply to work for the company online (see References).
Submit your work to Hallmark during one of the company's regularly scheduled themed contests. Hallmark does accept specific material on a regular basis for its contests. The winner does get paid for his entry, plus his work is in the hands of Hallmark. Although it is not a guaranteed way to get employment with Hallmark or to sell more of your poetry to the company, winning the contest does get your material in Hallmark's hands and can lead to employment with the company if your work is exceptional.
Follow the submission guidelines for the contest (see References). Not only will this increase your chance of winning the contest, but it also will demonstrate to Hallmark that you can write to specific assignments.