The OED (1919) has a 1775 sense for the noun stump as a speaking platform and 1814 for on the stump. It is, of course, from using a tree stump as a platform. It’s an Americanism, but the OED has this note: “In the U.S. the word ‘does not necessarily convey a derogatory implication’ ( Cent. Dict.). In Britain, though now common, it is still felt to be somewhat undignified.”
[Addition: Safire’s Political Dictionary has a 1716 reference to a stump being used as a speaking platform, but the word isn’t being used metaphorically. It’s a literal stump. Reading it, there’s no sense that the word was a term of art this early.]
The British National Corpus (1980s–1993) has several hits for on the stump, but it appears to be much less common than in the US.
There’s also stump speech, a standard oration that a candidate repeatedly delivers on the campaign trail.