Imagine a loop of pipe filled with water, and a pump pushing the water: The water moves around the pipe. The movement doesn't cause any part of the pipe to get heavier.
Imagine a different scenario, where the pipe ends in a big box: This time the box does fill up with water, and gets heavier.
Mapping the analogy from water back to electrons: a loop of pipe is like a loop of wire and a battery; while the pipe ending with a box becomes a capacitor or antenna, and that will leak[0] before you can measure the mass change — but technically yes the the mass of any given wall of a capacitor or of an antenna will be very slightly changed by this sort of thing.
For a sense of scale, to get a total charge of 1 coulomb using electrons, the mass of those electrons will be about 5.7 nanograms, and trying to squeeze that much charge into the last millimetre of some length of a wire 1mm in cross section diameter, involves about 60% of the energy in this explosion: https://www.youtube.com/watch?v=wqKn_3iJOP4
As nothing gets close to being able to hold that kind of energy, even if you're trying to accumulate a lot of excess electrons, those electrons leak well before even coming close to nanograms of excess mass.
Imagine a different scenario, where the pipe ends in a big box: This time the box does fill up with water, and gets heavier.
Mapping the analogy from water back to electrons: a loop of pipe is like a loop of wire and a battery; while the pipe ending with a box becomes a capacitor or antenna, and that will leak[0] before you can measure the mass change — but technically yes the the mass of any given wall of a capacitor or of an antenna will be very slightly changed by this sort of thing.
For a sense of scale, to get a total charge of 1 coulomb using electrons, the mass of those electrons will be about 5.7 nanograms, and trying to squeeze that much charge into the last millimetre of some length of a wire 1mm in cross section diameter, involves about 60% of the energy in this explosion: https://www.youtube.com/watch?v=wqKn_3iJOP4
As nothing gets close to being able to hold that kind of energy, even if you're trying to accumulate a lot of excess electrons, those electrons leak well before even coming close to nanograms of excess mass.
[0] The bit on the top with the sparks coming out is a capacitive electrode: https://en.wikipedia.org/wiki/Tesla_coil