The firms I mentioned are generally those who advertise in Museum News, Aviso, and set up booths at regional and national museum conferences. Do a search on the Internet using such keywords as museum exhibits, exhibit fabrication, museum contractors, museum services, museum exhibit designers, etc. The AAM also publishes a fairly comprehensive guide to museum services and suppliers as Vol 2 of the Museum Directory.
To get an idea of what types of work these firms do, take a look at
http://www.swmuseum.com
http://www.museumservices.com/
http://www.chasestudio.com/
http://www.jackrouse.com/
(There are many more, and I don't mean to be prejudicial, but these are just some I know of).

I sent a letter cold to the place I am at now. They had advertised several months earlier for a designer, and I kept the address. Then, when the time came, I sent a resume and cover, and within two weeks was starting a new job and back in the museum field where I wanted to be. 
Good luck,
Bryan

Wendy wrote:

 I found this very interesting, however (my apologies for not including who that was), I don't know what businesses these are, or how to find them.  Can anyone give me some leads?  Thank you.