That would depend if the chariot was wargear or a dedicated transport. I know in the WD update the Herald could not leave the chariot, so they were essentially the same. In that case I would say the re-roll affects the chariot too. But I've not checked the new codex that thuroughly. If the Herald can leave the chariot it would be a definite no. Unless, of course, the rules stated that the Herald's abilities also affected their chariot.
"Fetch me another plaything. This one seems to have broken." -Urien Rakarth
Space Marine- "This planet is ours, witch."
Eldar- "No ... This planet is theirs ..."