Hello, is it possible to use a crystal radio to run a horn speaker?

Not without adding an amp.

The horn speaker is 1275 ohms. Do I need a special amp?

If you have a strong local station it's possible to have a crystal radio with an amp that doesn't need a battery or AC supply. Some crystal radio guys build 2 radios.... One harvests a few volts of power from the local high power station to power an amp or regenerative stage and the other tunes the weaker stations for the gain stage.

What kind of amplifier without power. I am guessing a transformer? If so how do I find one?