Sorry for the delay in answering.
Thanks for the tip Miyakoji, I had asked at Uguisudani Station and was told in Japanese to get something at a bookstore but went under the impression that it was just blank sheets of paper. Bill937, thanks, actually that kind of stamp rally was what got me interested last year.
When I went on the weekend trip two weeks ago, I noticed a stamp and pad outside the turnstiles at Kiyosato Station on the Koumi Line and again at Ueda Station but only Shinonoi Station had paper provided for stamps. It had a logo character on the sheet, too.
Stamp pad stand at Ueno Station.





Some Summer Stamp Rallies.
Tetudo Stamp Rally. Visit 40 stations or Facility Open Houses outside of Tokyo and collect their stamps. Plus five more needed. Apparently you must use stamps from at least twi different rail lines.

Booklet with info and spaces for each stamp.

Tetsudo Stamp Rally.
Visit some stations and facilities or ride some trains (the stamp and pads are in the trains) to get your stamps.

Affix stamps to spaces in this brochure.

These are two photos of the stamp in Ueno Station.


There is also a trading card rally. There are 25 cards. 10 can only be gotten by patronizing participating restaurants (one to three per station) in stations in Tokyo. One card per station. The other 15 can be bought in two card packs, (100 yen each) random cards in each pack. But 8 cards can only be bought in selected NewDay stores in the western half of the Yamanote Line and the remaining 7 in selected NewDay stores on the eastern half.

Stamp Rallies at Ueno, Tama and Inokashira Zoos and the Kasai Aquarium.

Pokeman Stamp Rally.

Not every Yamanote Station has a stamp. There should also be Pokemon decorated Yamanote train, if last year is any guide.

Best wishes,
Grant