As far as I know Bungie-esque "vaulting" isn't a thing at all, at least outside of S1 which was never intended to be permanent/replayable content and was handled very differently (they were still releasing updates every 2 weeks, which was nuts).
Core story should sorta kinda go straight into LS S1, then you've got S2 which leads right into HoT. I think S3 is HoT, and S4 is PoF, with Icebrood Saga being the "it's like an expansion!" non-numbered season leading into EoD.
I'm very curious to see where the story picks up in EoD. Still need to finish these last few episodes (almost done with E4 and the bloody tower, at least everything it much easier than I remember now that I have a functional build with proper gear) and then get back to taking my engineer through HoT to get the final chievos while I replay through everything on that character for some achievements and shit. Mechromancer is still hilariously OP and makes everything super easy, but man is holosmith still my much preferred build. Really like managing the heat mechanic, just wish there was a better offhand weapon for damage than the shield (nice CC, but otherwise meh)