About 20 years ago, I created a community service project that involved giving new books to poor kids. It eventually was adopted by organizations in at least 17 states.
What I think helps is coordinating with a corporation that would benefit from your project, which is what I did by partnering with newspaper companies and bookstores. What comes to mind in your situation is crafts stores. They would benefit because people making crafts would buy the materials at their stores. Meanwhile, the crafts stores could host gatherings and training sessions for people who wish to have their organizations make the cards.
The store also may be willing to fund a brochure that would include how to put together the cards, and where to send them.
You also may wish to make Happy Cards a 501c3 nonprofit. That, however, takes about a year to do. You need to create a board and do other things. You can Google to find out info.