Another simple approach is to buy a cheap Pixie transceiver kit (about $5) and convert it to an AM transmitter.
This makes use of an LM386 as a modulator for a transistor power amplifier which is driven by a crystal oscillator.
It's easiest if done on 7 MHz (the Pixie design frequency)
But other frequencies eg 1.8 MHz are also possible with some mods.
Both are on amateur frequencies but have an inbuilt low pass filter. The 7 MHz version was faintly heard several hundred km away (when connected to a good antenna)