Hello Cooler King,
The tone on Immigrant Song is LP #1 through some amp with a solid state rectifier, possibly a Marshall or a Hiwatt. Page was using Hiwatts for his live rig so it's possible - just without the built in overdrive engaged. I nail that sound with my LP through either my Marshall or my '70s Hiwatt DR103. I say those amps because of the midrange coming through. Page was big on keeping the tone pure - capturing what sound was in the room rather than monkeying with EQ on the board - so the tone is coming from the guitar and amp. FWIW the Hiwatt is little more midrange than the Marshall; something to consider.
For the verses, it sounds double with another guitar run through a Leslie - that is where the vibrato sound comes from. The Plate reverb guess is very possible. For pedals, I have the Strymon Lex as a Leslie sim and the Catalinbread Talisman for plate - best plate reverb I have used, ever, and that includes lots of expensive rack gear. The Lex is dynamite and I fully recommend one to anyone looking to add that sound to their rig.
I hope this info is useful.