I can think of one huge organization that is funded by private individuals. National Geographic and their Geneographic project. You buy the expensive genetic test kit and it pays for their research... which includes digs and genetic testing of ancient bones. Many times organizations and/or private individuals come up with an idea and fund an archeological dig. The Richard III society funded a dig to find his bones in the UK, however their money only went for the dig and reconstruction of his face.
Leicester University wanted to do DNA testing, MRIs and a bunch more, so they funded for more. Most likely they received government money and wrote a grant, as this was a King of England and public domain. Most digs are public domain. Most likely the university kicked in some money and the government gave them more money in the form of that grant. So, a group of people with an organization is one way, but that may not cut it.
Governments fund science projects and archeologists put in a proposal for how they would go about finding something of interest. The best ideas get grants from governments. My daughter has a PhD in science and I know how funding works in many areas of science. It varies, but much of it is the same. Universities do not usually fund scientists, as they want scientists to get the funding from governments and other agencies, private or not. Why? The university gets a cut of the money to maintain the university.
A professor with a lot of grant money can basically write their ticket to a tenure track at many universities. A professor who brings grant money to a university usually gets hired. However, many of the Ivy leagues like Harvard and Yale have money to burn and sometimes fund things they want, like a piece of artwork or a manuscript. Writing a grant can get into university politics and determine who they hire as a professor... at times. That's a dirty little secret my daughter found out. To think otherwise is to be naive for a PhD.
Paying to go on an archeological dig is another way they get funding from people, but that is not enough. All that does is pay to feed the person and a bit more. That does not fund the excavators and many other scientists needed, such as MRI people who take images of the bones, etc. It can expand into the need to use many scientists and even doctors getting involved to solve a mystery, like a geneticist.
Funding is usually from many sources, as it was for the Richard III dig. A lot of funding comes from governments who are interested in tourism. Governments also set aside scientific money each year in their budgets. Digging in Egypt was probably heavily funded by their government after private funding from the past had found much. After that the government is interested. The more you find, the more for display and the better the tourism.
Federal law can't make a private company pay for an archeological dig on private land or the excavation of it. If a developer happens across something, they don't pay and they can plow right through it in the US. If it is a known historic site on public land, they pay for a survey in the US. Usually developers work around the area of interest and don't pay for the archeological digs. It's cheaper for them to work around the area of interest than get involved in paying for the excavation and more. On public land they have to have an archeologist do a survey.
It depends, but normally, universities want the archeologist or scientists to bring in the grant money from outside. The university gets a cut of that money, as I mentioned. My daughter writes a lot of grants. Some worth a few million dollars. One professor she knows has nearly 8 million in grant money from the US government, who sets aside a certain amount each year. The university immediately made him a tenured professor at the age of 32 and that is unheard of.