https://rechat.twitch.tv/rechat-messages?start=TIMESTAMP&video_id=VOD_ID
example
https://rechat.twitch.tv/rechat-messages?start=1458316988&video_id=v55153756
TIMESTAMP is the absolute epoch timestamp to start querying for messages. It appears each query will return a 30 second window of messages. This timestamp appears to delayed ~14 seconds from the true timestamp of the message likely do account for HLS delay.
VOD_ID is the ID of the VOD where the message was sent.
As always, this API is unsupported and not intended for 3rd party use so it might change without warning.