It varies by company, some use free lancers some write in house. Some want a hard copy some will take electronic submissions, etc. Pick up a copy of The Writers Market for the new year, it will give you step by step instructions as well as telling you who is in the market, if and how they accept submissions, what they pay ( probably no more than 25 dollars per item purchased,) how to approach them and the types of cards each company is liable to purchase - funny, romantic, dirty, etc.
This is the website
http://www.hallmark.com/online/careers/
for information on how to be part of the 'creative' team at hallmark including the application process.
I would say that most other companies would also need to see a portfolio of some sort.
Check out how to apply for Hallmark cards and that would give you an idea of what you need if you approach other greeting card companies.
Good luck, hope I get to see some of your poems in store.