But you wouldn't be voting in that case. It's no different than a Jury Summons. Maybe serving in a Jury is against your religion, but that doesn't mean you can simply throw away the summons or ignore it. No, you must follow the instructions, and either call, write, or show up at the required place at the required time. Then you can explain that serving in a Jury is against your religion, and hopefully the Judge would let you get out of it.
I'd imagine it would be an issue in the Jim Crow South, which ran on preventing blacks from voting.
That's why I don't think something like this could be ratified back then. There would be a solid block of southern states against it. You'd need every state except the ex-confederate sates +1 in order to pass it. If say both Kentucky and Oklahoma sided with the south then it would be toast.