You need to have the Green Book and if the owner isn't with the vehicle, you need a rental/lease agreement as well as copies of the owner's ID card.
Then you'll need a Thai "temporary export" form from Customs at the Border (TM2 Form (Temporary Export/Import Goods Form).
You'll likely need a similar form on the Lao side of the border as well.
It's basically the same as when we ride our motorcycles to Cambodia (or used to do).
Lots of copies of your PASSPORT. Customs will NOT care if you have a driver's license from an ASEAN country.
Make copies of the Green Book. Copies of the MIL's ID card. Copies of the TM2 after it's been approved. Copies of the Laos Customs document.
(GOOGLE what THEY want yourself.)
And it will NOT be a quick process even if you have ALL the paperwork in hand.
When I went to Cambodia with a Swedish friend and 2 Thai friends, we ended up spending nearly 4 hours at the border.
The Swede was on "his" BMW motorcycle - but the green book was still in the name of one of the Thai guys with us who was on his Harley.
They had to go get a "rental" contract and a "postal stamp" (no lie, an actual postage stamp for one of the documents). Took us about 2
***
hours on the Thai side to get everything done.
And another 1
***
hours on the Cambodian side because we had to go from the "Immigration" booth to the "Customs" booth, then register online on the Cambodia Customs site, fill out an online form and then the Customs guy printed it out and stamped it and then it was back to the Immigration booth.
Easier on the way out. Stopped at the Thai Customs checkpoint, rode over to the Customs checkpoint, handed in the copies of our paperwork, got stamped and on our way in about an hour.
The same friends (minus me as my bike was in the shop) went to Laos last year.
I recall they said it was a fair wait at that border as well so be prepared.