No. In the US a mailbox is for mail with postage, and that covers the inside, the door and the post-mounted box itself.
The short answer
No, not without postage. Under United States Postal Service rules, a mailbox may be used only for matter bearing postage, so a lost pet flyer you slip inside, clip to the door or tape to the box breaks the rule even though you mean well. The good news is that the rule only covers the mail receptacle. You can still hand flyers to people, leave them at a front door where that is allowed, or pay postage and mail them. The fastest legal way to reach a street is usually to knock, hand over a flyer and ask one specific question about sheds, garages and cameras.
What the rules actually say
Two sources matter. The USPS Domestic Mail Manual, section 508.3.1.3, says a mail receptacle may be used only for matter bearing postage, and that no part of a receptacle may be used to deliver matter without postage. It spells out that this includes items placed upon, supported by, attached to, hung from or inserted into the box. Federal law, 18 U.S.C. 1725, separately covers anyone who knowingly and willfully deposits unpaid mailable matter, such as circulars, in a letter box on a mail route with intent to avoid postage, and it provides for a fine. We are not lawyers and this is not legal advice, but the practical reading is simple: keep unstamped paper off the box.
What counts as part of the mailbox
People often assume the rule stops at the inside of the box. It does not. The manual’s wording covers something placed on top, attached to the side or hanging from the box. That rules out a flyer tucked under the flag, rubber-banded to the door, or taped to the front. Cluster boxes in apartment complexes and subdivisions are mail receptacles too, so they are off limits for the same reason. Ask the property office instead. Many complexes have a notice board or will email residents, and that often reaches more people than a stack of paper would.
Door slots are treated differently
The manual says door slots, and nonlockable bins or troughs used with apartment house mailboxes, are not letterboxes under 18 U.S.C. 1725. That does not give you permission to push paper through every door in a neighborhood. Check for a no-solicitation sign, respect it, and stay off anyone’s property when they have asked you to. If your city or homeowners association restricts door-to-door flyers, follow that rule as well. When you do leave a flyer at a door, wedge it where wind will not carry it into the street. Folding it into the gap of a screen door usually works better than leaving it on the mat.
Better ways to reach every house
Walk the street in the evening when people are home. A short script works: I’m looking for my cat, she went missing Tuesday near Oak and 5th, would you check your garage and shed tonight? Hand over the flyer while you say it. If nobody answers, leave it at the door and move on. Use the two-per-page flyer so the same sheet of paper covers two houses. Keep a simple list of streets you have finished so helpers do not repeat them.
Talk to your mail carrier
Your carrier cannot deliver unstamped flyers, but they cover every house on the route, and they see yards and porches at hours you do not. Hand them a flyer directly and ask them to keep an eye out. Do the same with delivery drivers, landscapers, school crossing guards and anyone else who works outdoors in your neighborhood. These are people who pay attention to the street for a living, and one flyer in a truck can cover more ground than a dozen on a fence.
If you want to mail them
A stamped, addressed flyer is ordinary mail, so it can go in a mailbox like anything else. This only makes sense in a long search or for a small set of houses backing onto the place your pet was last seen, because it takes longer than walking and costs more. Fold and seal the flyer, address it to Resident at the actual street address, and pay the correct postage at the post office. For the first days, people on foot will beat the mail every time.
Sources
USPS Domestic Mail Manual 508, sections 3.1.2 and 3.1.3: pe.usps.com/text/dmm300/508.htm. 18 U.S.C. 1725: law.cornell.edu/uscode/text/18/1725.