Geography, at a first pass, over multiple elections. This is how TV testing works. Pick "similar" geographies and run your marketing in one. If the effect is large enough, it pops out. Not quite a diff-in-diff, but a start.

Or don't look at votes, look at candidate likes and shares over time, especially as they shift.

The defined metric doesn't have to be "propensity for this individual to vote for a candidate." It can be "percentage delta over untreated markets compared to prior campaigns."

Maybe with polling?

Data is terrible, especially for polarizing candidates like Trump. People simply lie in public about not voting for him, afraid of backlash that they will receive.

They were going to vote anyway; nothing can tell you otherwise?