Sexual intercourse, especially that of a casual and saucy nature.
This reduplicated term sounds as though it might come from one of Shakespeare's more baudy scenes, but is in fact of quite recent coinage. I can't find any mention of it in print prior to the entry in the 1968 Scottish National Dictionary:
Rump, Rumpie-pumpie, a jocular term for copulation.
We don't know who coined the term. Whoever it was didn't need to look far for the inspiration though. The comedy radio show Round The Horne was a a highly popular in the late 1960s. Barry Took and Marty Feldman created the character of Rambling Syd Rumpo for Kenneth Williams and each week he would sing a stream of delightfully nonsensical innuendo, usually to an authentic English folk melody; for example, The Highland Lament - a song of the Highland Turv-croppers, from April 1967:
Ye nurves and bogles of Bonnie Glen Postule
Oft have I greebled amongst your trees
And whirdled my lassie among your nettles
I swore my devotion and stung both my knees
For we bogled and grundled and grundled and bogled
I grottled my muckles for bonnie wee June
And now she's run off with a plumber from Sidcup
So I lurk in the forest and whirdle alone
The use of suggestive comic gobbledegook was used extensively by radio scriptwriters, notably Took/Feldman and Spike Milligan, as a device to sneak 'unsuitable' content past the BBC censor. The sexual imagery of 'rump' and 'pump' leaves little to the imagination and, whilst Rambling Syd Rumpo didn't explicitly list rumpy-pumpy as one of his pastimes, it isn't difficult to imagine that it would have been.
See other reduplicated phrases.