What is the most economical way of installing an up-to-code emergency phone system into our church’s small elevator that is seldom used and travels about 12 vertical feet between two floors? We’d like to invest the smallest amount for both installation of a system, and for its monitoring as it appears that this is an ongoing requirement, as well. Any similar experience or advise would be appreciated.

Most of the mobile towers have no network coverage there in underground chambers, tunnels and cellars. On the other hand, in the lift, the tower fails to catch the link as the lift acts more or less like a "Faraday cage" or even if it does catch the link, the phone conversation gets severely disrupted.

Quoting from the wikipedia link by pela in the comment below,

Elevators and other rooms with metallic conducting frames and walls simulate a Faraday cage effect, leading to a loss of signal and "dead zones" for users of cellular phones, radios, and other electronic devices that require external electromagnetic signals. During training firemen and other first responders are cautioned that their two-way radios will probably not work inside elevator cars and to make allowances for that. Small, physical Faraday cages are used by electronics engineers during equipment testing to simulate such an environment to make sure that the device gracefully handles these conditions.

EDIT:In addition to what I have written about the communication failure in cellars, I am including the following:

Connections are simply electromagnetic radio waves and you won't get reception where it's impossible for a wave to pass by to. So, in basements, most waves get reflected and absorbed at many levels and also a bit attenuated, thus you can't get a proper reception over there.

Also it can be noted that whether signals can penetrate into tunnels,cellars, etc. is a function of how well they can diffract compared to the size of the corresponding structures.

From this wikipedia page on mobile signals, it can be quoted that:

Underground areas, such as tunnels and subway stations, will lack reception unless they are wired for cell signals.

which resonates with the first line of my answer.