Welcome to the forum. I'm not sure where you could find a 'new' board. I don't think replacement parts are typically available for these types of props - at least not OEM. Might possibly find a rare one on ebay or something, but I'd expect that to still be a long shot too. As far as 'parts' - if you mean individual electronic components, there are lots of places to get those... digikey (personal favorite for easy searching), mouser, newark, ebay, likely others, too.
Have you taken anything apart to look inside? If you do, take some high quality photos of the board and possibly some eagle-eyed members might spot something. One thing I typically have a lot of trouble with is the main wires breaking loose from the battery or power socket. I'm not sure how they do it sitting untouched in storage, but seems like ever year I have to solder a couple of wires back onto their tabs because they've broken. Big electrolytic capacitors are also a 'usual suspect'. Sometimes in a hot attic or storage shed they can loose the electrolyte and short out. Once that happens, then you have a direct short across the power supply/battery and the whole rig won't work.
As far as troubleshooting - does it have a battery supply? if so you might try that as an alternate source of power - or if using batteries now, maybe change it over to a wall wart / power supply. Do you have the ability to measure current draw? If so is it drawing any current? Is there any on-off switch? Sometimes those need a lot of back/forth switches or jiggling to work oxide off the contacts and get a prop working again. If you get a look at the circuit board, any visible damage? Water stains/corrosion? any burned or obviously 'hot' components? damaged circuit board traces? again, good photos would help here.
If all else fails, other ways to get it working - certainly many out there - anywhere from the very low end of possibly just running power to some motors, relays and leds to some sort of arduino sketch all the way up to some sort of 'brain box' like an octobanger or even a monsterguts control. All depends on how far you want to take it.
But let us know on some of the questions above and maybe we can get him dancing again.