PEX, and be careful with it.
Are those flex supplies one piece to the valves....the ones leading to the faucet.......?
If they are not, then you could possibly put a add-a-valve to that equation without buying expensive crimpers and pray the rented ones are calibrated correctly.
Otherwise it's call a plumber time, or get sharkbites and make up a tee connection to feed the RO system. You'll have trouble finding something to bring you down to 1/4" though in regards to sharkbites.