Why Banks?
The United States Mint directly deposits coins to all American banks as reserves. At the same time, the United States Mint periodically releases coins for collectible reasons. For example, during the 200th birthday of Abraham Lincoln in 2009, the Mint released to all banks special Lincoln pennies that now serve as collectible items.
Large Banks
Larger, national banks may seem distant, but all their local branches can participate in a coin collecting program for children or distribute out rare coins. Large banks have an advantage of utilizing large reserves from the US Mint concerning collectible coins.
Local Banks
Non-national banks are also prime locations for a coin collection program. Local banks in your area might have rare coins available within their vaults. Local banks may also be more willing to work with school or community programs that educate or organize coin collecting circles.
FDIC
If you are really having a difficult time finding local or large banks with coin collection programs or collectible coins, you can contact a regional Federal Deposit Insurance Corporation (FDIC) office. The FDIC office can network you to the banks in your region that do have coin programs.