I would not say they are popular, but you can get what might loosely be termed “souvenir passport” stamps, because they are commonly provided as part of a promotion which uses the conceit of a passport to record visits. By the same token, not every promotion or package billed as a “passport” involves any physical stamps, however, and some programs now only offer electronic stamps.
The most popular souvenir passport programs, I would surmise, are the National Park Passport (officially the Passport To Your National Parks®, but no one calls it that), and the Major League Baseball Pass-Port. In both cases, you can obtain a stamp from the location you visit for no additional cost beyond admission, and regardless of whether you are enrolled in the visitor program or using their “official” booklet. The stamp and inkpad are commonly available at a counter, and you stamp your own. At a minor league ballpark, you may need to ask more than one person where to go.
Should your visits include a presidential library or a historic lighthouse, you might also look at the Passport to Presidential Libraries, sponsored by the National Archives, and the Lighthouse Passport Program sponsored by the U.S. Lighthouse Society, which should also be available without charge or enrollment.
A more obscure program known as the Blue Goose Passport is sponsored by the U.S. Fish and Wildlife Service, if you are visiting locations within the National Wildlife Refuge System. Again, I find it very unlikely anyone will notice or care if you do not use the official booklet.
There is also the Appalachian Trail Passport, aimed at hikers who want to keep a stamp book as a memento of their trek. Unlike the previously mentioned programs, the stamping locations are mostly private businesses, where at least a token purchase might be polite.
Beyond this, there are various local or regional promotions which may be organized as a “passport” program. Kentucky offers stamps on its Bourbon Trail and New Jersey at some of its wineries, for example.
As a last option you could try for a postmark from the local U.S. post office. You could purchase a stamp and ask them to cancel it in your notebook to collect the local postmark as a souvenir. This is likely to be an unusual request at the vast majority of post offices, so good luck if you want to try.