Raining Blood by Slayer.
Hard to believe no one has mentioned the 4th movement of Beethoven’s Symphony #6 (“Pastoral”), memorably illustrated in Disney’s FANTASIA. (Oh, wait, sorry, I see you said “with vocals.”)
In The Rain – The Dramatics
Paul McCartney’s Uncle Albert/Admiral Halsey, around about the lyrics “and I believe I’m gonna rain”.
Storms in Africa, Enya
There’s Rain by the Beatles, although it’s just a reference w/o any actual pitter patter.
Do any versions of Toto’s Africa have storm sounds? I can’t recall. The lyrics (actually the chorus) repeatedly mentions “the rains in Africa.”
“Rainy Day In June” by the Kinks
“Short Change Hero” - The Heavy
There’s a thunder roll following the too-long intro before the actual song.
Steelhouse Lane - Son of a Loaded Gun
Cat Power - Say
Sting also has Wild Wild Sea from The Soul Cages album. A long time ago, when that album was still fairly new I remember I was out walking the dog listening to the album on a Walkman on a cloudy day and I thought the thunder was real. 
Both of Tomoko Kawase’s Halloween singles (2006’s Lollipop Candy BAD Girl and 2011’s I’m Your Devil - both released in her Tommy heavenly[sup]6[/sup] identity), include the sound of a thunderstorm over the intro and outro. It doesn’t overlap with the vocals in LCBG - IYD, it’s part of a spoken intro/outro - but, hey, I can’t pass up a chance to mention Tommy.
Cee Lo’s “Bright Lights Bigger City” has a police siren in it - the first two times I heard the song I was driving, and looked around to see if I ought to pull off the road!
Matthew Sweet - “Thunderstorm”
Clouds The Rocket
Yes South Side of the Sky
Surprisingly, The Sky is Crying (SRV) does not.
Cars Hiss by My Window by The Doors - think it has rain sfx in it.
Ha! I once partied in the dumpy little hotel room that this was written in!