Can you be a postdoc with funding from a private organization (vs. government funding)? Can you be a post-doc if you secured funding from a private organization to do research at a university in the social sciences/humanities, or do you have to have governmental funding to do a postdoc?
It is absolutely possible to have a postdoc funded by a private organization - there's nothing about a postdoc inherently tied to a particular type of funding. A couple ways this might happen:


As @Dexter mentions, there are private trusts and charitable organizations that fund research. This can occur either by funding a fellowship directly, or allowing funds from a grant to be used to hire a postdoc.
There are also commercial companies that have research grant programs in a number of fields. It's entirely possible to use the funding from one of these to pay for a postdoc.
A private university could directly pay for a postdoc - for example, as part of a professor's startup package.