I read better part of the thread before noticing how old it is. However, about age of Ford racing team. If we believe wikipedia it was founded in 1901.
Ford Performance - Wikipedia
Given that in that period, the "Ford" mentioned was not the Ford Motor Company of today, and if anything, that would actually be Cadillac. So, Ford Performance definitely could not be stated as being founded in 1901. Henry Ford was racing his stuff, but that isn't the same thing as having a specific team entered in sanctioned competition on a continuous basis.
Similarly, Mercedes-Benz has entered cars in competition ever since the very first motor race of any kind. But it hasn't been on a continuous basis with one organization, so it can't be counted either.
I think Ferrari comfortably takes this one, with probably Porsche, and Honda being the next oldest options. As a non-manufacturer, private team, the Wood Brothers are probably the correct answer, I can't think of anyone else other than Penske who could be a potential rival.