Do I need a re-entry permit for a one-year multiple entry OA visa when leaving Thailand?

You will stamped for a year on a Non-O A multi entry. If you get single entry it 90 days. As for the CoE I myself personally got it as well as the visa. It took one day on the online portal

Okay the way to tell IF you need a re-entry permit or not BEFORE you leave the country IF you have a year-long, multi-entry Non-Immigrant Type O-A visa is to LOOK at the visa sticker <- that you got from the thai consulate in your country before you came here and see what the "enter before" OR "valid until" date is on it

IF you are going to return to thailand BEFORE that date you do not need a re-entry permit and as long as you have the required health insurance you will be stamped in for a year (or as long as the insurance is valid up to a year)

Here is an example of a year long multi-entry O-A visa issued from Canberra on the 18th of Oct 2018 and valid until the 17th of Oct 2019

Now as long as the holder of that visa re-entered the country on or before the 17th of Oct 2019 he would have been stamped in for a whole new year

Its multi entry why would you need a permit?

Nope

"Multiple Entry" is the clue..

As long as your visa is in date when you return you won’t need to renew it but you will need to reapply for a COE to enter Thailand once you leave.

No