Do you need a special permit to hand out flyers in people's mailboxes?

Legally you are not allowed to put anything in or on a mailbox unless you are an authorized U.S. Postal Carrier.

From straightdope.com, "[An] example of the federal interest in your mailbox is the limitation placed on its use. DMM 508 section 3.1.3 says a mailbox "may be used only for matter bearing postage." It's a federal crime to deposit mailable matter without postage into a letterbox, and many items are officially nonmailable. So, Stan, the answer to your question is yes - you can't put anything in a mailbox that doesn't have postage on it. For better or worse, neither can all those guys coming around with advertising flyers, which is why you get stuff rubber-banded to your doorknob or scattered on the lawn.

The idea behind these rules is to give the postal service a monopoly on mail, a privilege formally conferred by the Private Express Statutes. According to DMM 608, the statutes prohibit transmitting letters or packets by non-U.S. mail means without paying postage. There are exceptions, of course. The one that gets the express delivery services off the hook permits private delivery of "extremely urgent" letters. Quality Service Guide 608 spells out what qualifies as "extremely urgent" (letters sent from less than 50 miles and before noon have to be delivered by the end of the business day, for example), but there's an easier way: "It will be conclusively presumed that a letter is extremely urgent … if the amount paid for private carriage of the letter is at least three dollars or twice the applicable U.S. postage for First-Class Mail (including priority mail) whichever is the greater," says 39 USC 320.6. In other Words, if the express delivery service charges you enough, the postal service will assume your letter or package really did have to get there overnight. On the other hand, if the item fits the statutory definition of mail and isn't covered by one of the exceptions, it can't be delivered. In Associated Third Class Mail Users v. USPS, the Court of Appeals ruled that a group of advertisers couldn't develop a private system to deliver generic advertisements - it had to send them through the USPS."