Are there archeology companies? Or do they often work for universities and get research grants of some sort?
Real archeology almost always involves research grants from federal, state, and maybe private sources. Good archeology isn’t very compatible with a “for profit” model even though I am sure you could find an example that is. However, there are some big money salvage operations, basically treasure hunts, that operate off of venture capital and often search for shipwrecks from the Titanic to Spanish gold ships and some of these have paid off in a big way.
There are archaeolgical companies. Many times what happens is that a new road or building is being put in. If someone happens to find something they are supposed to stop work and bring in someone to evaluate what they’ve found. This of course depends on the location as some places are a lot less strict then others. These companies will also do a lot of surveying and other cartographic work. I had an interview once with a Dr. Jones.
Then there are the universities that have departments. A lot of times they will find a place and do a dig for a couple of years using the students. They use a lot of volunteers. There are also organizations that will help out with these digs. I would say every US state has a state organization and smaller county ones as well. I was a member of the Maryland group for a few years. People actually pay to help out as a way to raise money.
Governments also have small groups as well. They tend to do the same type of evaluations that the companies do when there is work done on public land.
On the few digs that I have done I’ve worked with American University, the Maryland Archaeological society and a couple of state workers.
Mostly, aren’t archaeologists either university faculty members, or else they are employees of some major museum (e.g. the American Museum of Natural History, or the Field Museum, or maybe the NHM of L.A. County). I’m thinking here mainly of the fossil hunters, and not the seeker of artefacts from ancient civilizations.
Those would be paleontologists.
First, they must hit “pay dirt”… 
IT BELONGS IN A MUSEUM!!
Some countries have more state-sponsored serious archaeology going on than the U.S. does. For example, in Mexico, many serious, peer-review-journal-published archaeologists do not work for universities, but are rather direct employees of the federal archaeology and history institute (INAH). Some do both. And, as in the U.S., many full-time archaeology professors also depend partly on grants from the state insititution.
A friend of mine used to work for such a firm in Georgia. He now works for the NC Highway Department.
Basically, before any road goes in, he goes and digs test holes [1], and makes sure they don’t pave over any significant sites. Oddly enough, he doesn’t encounter as many Nazis as you might expect.
I’d also had a look at his field kit: there isn’t a bullwhip to be found. What a ripoff.
[1] I’m sure there’s a technical name for it.
I know a chap who works for the Museum of London Archeological Service. London is of course mostly built on London, so pretty much any digging turns up something and archeologists seem to be interested in practically everything. One of their tasks is simply to hang around* at a building site to see if they uncover anything.
*OK, maybe it’s a bit more active than that.
So do you! Throw him over the side.
Stranger
A friend of mine worked at one of the Jamestown sites while he was getting his Master’s in Archaeology. He now works for a private foundation, which when explained to me sounds a lot like one of Gfactor’s CRMs. He is working on a site right now, searching for and categorizing artifacts from the Civil War era on what was then a plantation. He is paid by the foundation.
It could be the APVA (Association for the Preservation of Virginia Antiquities). They have a few archeologists on staff.
As has been pointed out, some engineering firms have them on staff as well to do cultural resource assessments of sites. There are also a few private firms that do nothing but those sorts of investigations, but I think they’re pretty rare. The likelihood of there being such a firm may have something to do with the area in which you live. Here in VA, with a pretty rich history, a Phase I or Phase II archeolgical study is required fairly often.