9 speed chain should work fine; but why not just use 8 speed (e.g. SRAM PC870), which are still widely available?
That wasn't the OP's question. He wanted to know if a 9 or 10-speed compact
crank would work with an 8-speed cassette and chain.
Yes it should. You may have some rubbing of the chain on the inside of the large chainring when you are in the small chainring and the smaller rear cogs. A 9-speed chain will reduce this problem and will work with your 8-speed cassette.