The search form in zenphoto documentation has no button.
http://docs.zenphoto.org/

No, but enter does work. Sorry, that's an automatically generated layout we use as provided.

Enter doesn't work by me. (latest Firefox)

Works for me with drop down of suggestions in latest Firefox, Safari 11 and latest Chrome. Perhaps you have some script blocker?

I have seen. Works only with searchstrings are in the dropdown exist.

It is static html with some js that the doc generator creates so not a fully dynamic search as on the main site.