A purchased packet of seeds often contains more than one gardener needs, so you may be able to find someone in your neighborhood willing to share. The Dinner Garden, hoping to resurrect the idea of victory gardens, provides seeds to anyone who asks. This group will mail seeds anywhere in the U.S. and also partners with distributions sites in several states, where you can go and pick up seeds.
Also check with your local garden club. They may organize a seed swap, or would be willing to start one this year. Check the "free" listings in online sites, such as Freecycle.org or Craigslist. Traditional newspapers might be a better bet in a small town.
The America the Beautiful Fund, started in 1965, offers direction and support to community projects throughout the United States. The non-profit organization receives donated seeds and bulbs from nurseries and in turn gives them to community garden groups and others interested in making their communities more beautiful. To be considered for a grant, fill out the application on their Website.
A number of seed companies offer a free packet of seeds to customers who place an order. And Redwood City Seed Company will send two for kids, provided you follow the guidelines on their site.
Some places will send U.S. residents seeds if the request is accompanied by a self-addressed stamped envelope. Mow's Garden offers a free packet of flower seeds. Heirloom Tomatoes sends tomato seeds. Winter Sown, which collects donated seeds, will send six packets of seeds. What you get depends on what has been donated recently. For Winter Sown, follow the directions exactly as it also requires you to send a printout of its Web page.
Ed Hume Seeds participates in a "Plant a Row for the Hungry" program. Gardener Ed Hume annually offers a free packet of vegetable seeds to 250 people who are willing to donate their excess garden produce to a soup kitchen or food bank.