I am trying to find a short story (1970's or earlier) with a society where people all get one vote at birth but can accumulate additional votes based on education, military service, and other positive citizenship accomplishments.
You must understand, the constitution gave every man a vote; therefore that vote was a vested right, and could not be taken away. But the constitution did not say that certain individuals might not be given two votes, or ten! (…)
The new law was framed and passed. Under it every citizen, howsoever poor or ignorant, possessed one vote, so universal suffrage still reigned; but if a man possessed a good common-school education and no money, he had two votes; a high-school education gave him four; if he had property likewise, to the value of three thousand 'sacos,' he wielded one more vote; for every fifty thousand 'sacos' a man added to his property, he was entitled to another vote; a university education entitled a man to nine votes, even though he owned no property.
The general idea is known as plural voting and has been applied in a few places in the real world.
Twain's story does not mention military service. That may come from a Heinlein story. Heinlein famously made the vote subject to military or civil service in Starship Troopers, and he explicitly refered to Twain's essay at some point, but the voting system in Starship Troopers is not a plural voting system.