Use a simple dynamic microphone plugged into a simple audio mixer that allows multiple output for the audio. The Public Broadcast Cart uses an old, incredibly sturdy and handy SHURE mixer donated by Jan McLaughlin. The closest version available on the SHURE site is the FP23 Microphone Preamplifer
Ramsey Electronics offers a great selection of affordable small transmitters from a $35 AM Radio Transmitter Kit to $300 Factory Assembled and Tested Digital FM Stereo Transmitter. The Public Broadcast Cart uses Ramsey's Professional Synthesized FM Stereo Transmitter Kit, it's an excellent learning experience to assemble your own transmitter.
If you have a wifi equiped laptop and an open wifi node available all you need is Internet radio broadcasting software. Rogue Amoeba's Nicecast allows you to turn your own computer into the broadcasting server. A better idea is to use Nicecast to stream the audio to a server designed to handle online broadcasting.
For immediate area audio amplification, just about anything that can recieve audio from the mixer and output it through speakers could be used, even a boom box. The Public Broadcast Cart uses the Samson SERVO 120A 120-Watt Power Amplifier that powers car speakers that have been mounted onto goose neck microphone extensions
The Public Broadcast Cart has used a few different power sources from as poor as car batteries to as a stable power supply as the Galaxy Far Outlet Model 300S, which Gotham Sound rents for a very low price.
Tetsuo Kogawa's 1 Watt Transmitter Schematic: