Your best bet is to find out which contractor makes them (Thiokol, Alliant, Boeing?) and where (Magna UT, Huntsville AL, or the CA desert), and start filing Freedom of Information Act requests.
You can also use any of the excellent textbooks on solid rocket motors available at a college engineering library to come up with thrust, Isp, and burn time numbers based on the diameter and weight.
The inventor of the predecessor to the original JATO bottle was Jack Parsons; if you research his life (and try to weed out the bits about his Satanism) you’ll get some good starting points, especially on the JATOs we don’t use anymore.
Oh, and you wouldn’t happen to be trying to use physics to prove or disprove the oh-so-popular urban legend about the JATO car, would you?