I'm sure some one will have a more authoritative response but from what I recall reading his Jazz bass is strung with old Rotosound steels, his P Basses with La Bella Flats and his new MM signature has Ernie Ball cobalt flatwounds.
Amp wise I believe he uses a Markbass Little Mark Tube 800 through a 4x10 cabinet, though in some of the Vulfpeck YouTube videos he seems to play through an Ampeg Combo.
I think the majority of his tone comes from his technique and EQ. I have a Markbass LMT, a 4x10 cabinet and a P Bass with Labella flats and I sound nothing like Joe, even when playing his lines.