Vanilla Fudge and Bloodrock come to mind.
Runaway by Del Shannon – is that an organ or a keyboard? I have no idea. I don’t know the difference. If there is one. ??
Pink Floyd used organs on some tracks; I can’t recall to mind exactly whic ones, and whether they would actually qualify as ‘rock’.
… one reason taht Carl Palmer was well-suited to later oin ELP.
I had always thought that Watcher of the Skies started out with Mellotron… time to listen to it again I guess…
It’s not an organ, it’s an electronic keyboard called the Clavioline. (Geeky nitpickery: the keyboard player on “Runaway” modified his Clavioline, and renamed it the “musitron.”)
In the 60s and up to the mid-70s, pretty much everything had at least a Farfisa or a Hammond B3. Since then, well, who listens to that crap anyway?
(grabbing my cane and hobbling away)
Jethro Tull’s David Palmer used to play a tiny pipe organ onstage in the late '70s. It was based on the medieval “portative” organ, and can be heard on the Bursting Out live album as well as on studio albums of that period.
I just gotta say this - there is a huge difference between a keyboard only organ and a pipe organ. And anyone who thinks that a pipe organ is easy to play … well, think again: those pipes are “played” with the feet, at the same time the hands are playing the organ keyboard. There are many classical pieces with very complicated pedal work in them!
Sorry, I do not know any rock songs with pipe organ in them. Interesting note about Jethro Tull, though - I didn’t know that.
The opening of Supertramp’s “Bloody Well Right” is all organ - I think it’s a Wurlitzer 200.
Also, pretty much the entire Ray Charles catalog. “What I’d Say” is awesome.
Really, you could name hundreds of songs here. The organ is a staple of rock music - at least as much as the piano.
Santana’s Oye Como Va has a nice organ part.
Keith Emerson made his name playing organ for the Nice before ELP.
Al Kooper’s organ on “Blonde on Blonde” was very influential: he basically hung around the sessions begging to join in and finally was allowed to play organ on songs like “Stuck Inside of Mobile with the Memphis Blues Again.” It influenced Procul Harum and the organ became a legitimate rock instrument.
The Beatles made a nice use of organ in “It’s All Too Much.”
Lee Michaels has great rock and blues primarily with Hammond B3 and drums. He is definitely a flashback from the 60’s and 70’s.
Woe betide the man who thinks an organ is easy to play. 
If we’re talking about English prog rockers, though, not to nitpick, but organs in the style of the English cathedral seldom had a pedal board bigger than the little rack on a Hammond; you won’t find expansive pedal parts on Handel’s music, for instance. It mostly falls within an octave. But yes, your point is well taken, with registrations, various manuals, and just a mess of coordination.
I like the bluesy soul rock organ Mark Knopfler uses; think the opening to Dire Straits’ “walk of life,” or “Everybody pays to play.”
Slightly off topic, but a classical composer from Iceland by the name of Jon Leifs wrote an organ concerto in 1930:
Frankly, I think it sounds like proto-metal music, myself. Does anyone else hear it?
Foreplay and Smokin’, by Boston.
Edgar Winter - Frankenstein
Not “rock” but … there’s this…
Take a look at Brian Auger’s work. He was rock to begin with, especially his work with Julie Driscoll, but then wandered over to jazz.
And the organ is almost entirely responsible for the wonderful chill-inducing crescendo at the end of “The Musical Box”.
I saw Brian Auger last year - still great!
Here’s a great [little clip](and don’t forget www.transreal.co.uk and transrealshop on twitter!) from 1968 of him performing Black Cat. Great visuals, weird dancers!
Did the Zombies ever do any songs with an organ?
Oh, yes; essential part of their sound.
I saw their Oracle & Odessey anniversary tour in 2009 and Rod Argent (mentioned up-thread) played a modern keyboard but also the original organ he had used in the recording sessions back in the 60s.
I remember him saying it was a nightmare transporting it for the tour!