Do VCRs have service menus?
 

Do VCRs have service menus like TVs?

Many later ones have a "service mode" for stuff like adjusting head switch, freq response etc yeah. Varies a bit how much you can do though. JVC ones and sony video8 gear require a special service remote to change stuff, others like the later PAL panasonics, european philips decks etc have a service mode you can access with some button combination.

Do you know about any website/document which lists the VCR types (models) with their service codes?

I may not need to change anything in the menu, I just want to see it.

Details on any service menus should be covered in the factory service manuals.

You can find information on the web for some models. Sometimes a search on: <model> hacks will produce results.

FWIW: Many Sony (and Canon) Video8/Hi8camcorders and VCRs used a modified wired remote control (Sony RM-95) that connected via the LANC terminal to make adjustments.