The Watch takes no part, but the flip side of that deal is that while the noble houses and knights and lords and ladies may squabble amongst themselves, they're supposed to leave the Watch out of it, and further they're supposed to all be helping support the Watch.
Out of all of those squabbling contenders for the Iron Throne, the only one who could be bothered to do any more for the Watch than to tell them to eff themselves was Stannis.
So Jon bends the rules a little, lets Stannis take some old spears and junk armor which he could have taken by force if he'd been of a mind to, gives Stannis a little Northerly advice, because otherwise he's letting his only ally against the others ride out to be slaughtered by the Boltons.
And then comes the Pink Letter, which personally threatens Jon and more generally threatens the Watch as well.
So to protect the sworn brothers of the Watch Jon detaches himself to ride south without them to intercept the Boltons to keep them away from the Watch and The Wall.
It just so happens there're are a bunch of non-sworn wildlings who accept his invitation to ride along and keep him company.
Jon wasn't going to retake Winterfell, he was going to intercept the Boltons so as to keep all the rest of the Watch but himself out of the conflict. Winterfell is no longer his home.