How to sample tiles along a long-distance route

Hi folks, newbie here. I was looking for an OSM service that allows easy tiling along a long-distance hiking route. These tiles would have to be printed out for offline paper access and arranged in a way that uses as little paper as possible.

I didn't find such a service. Is there one?

If not, I would love to script a solution and share it. Do you have some quick hints on where and how to start with the OSM API to tackle this specific problem?

Edit: The API is not meant for read-only purposes (https://wiki.openstreetmap.org/wiki/Api.openstreetmap.org/Usage_policy), so I doubt that one can retrieve tile PDFs. If you have any idea how to do this without manually traversing the route and Share->PDF, please let me know.