Nonprofits may do basic research, but by definition they are not academic institutions like universities, and their missions are usually broader and more applied (e.g. funding, disseminating, and championing research applications) than doing basic research itself, especially if the nonprofit is relatively large.
Also, nonprofit work is not technically post-doc, as post-doc is typically a title reserved for work in university research center or lab settings, within the framework of a traditional academic institution.
There might be borderline instances if a nonprofit is very tightly integrated or affiliated with a university lab or center, but for the most part post-doc positions are only advertised by universities (at least in my experience).
As long as you are venturing outside the academe, I would encourage you to consider opportunities in government research labs or agencies, such as state departments of health. They provide solid research experience and can be a good 'staging area' between the academe and the private sector, for you to spend a few years and build your CV. Good luck!