So, here's the update:
(1) The RGB (VGA) + audio cable I ordered is doing the trick! I'm perfectly satisfied with the screen resolution I'm seeing and thrilled that this was such a cheap fix. Yeah!! ;D
(2) The repair people that Vizio referred me to no longer support my TV (it's only 3 years old!!). When I called Vizio back, they said that since they're no longer making this TV, they also don't have parts, so replacing the board that runs the HDMI ports depends entirely on whether a distributor still has it.
I guess this was my up-close-and-personal lesson in planned obsolescence? (An expensive 24" TV that's only 3 years old going straight to the dump??)
My biggest concern was that the board that ran the HDMI ports might also be the one that supported the RGB port. I am so relieved that this is not the case.