I got this on the Island map with the K-700, the large tractor type truck. If you go the first lumber mill right by the spawn and head northeast along the paved road, you’ll find it on the left side of the road. Hop in and head further northeast until you get to the Log Station.
You’ll have to create some logs at the Logs Kiosk since you can’t auto load logs into the claw on the back. I used short logs so there was less chance of them getting caught on trees. Once there are a couple logs in the trough, back up your K-700 to them and use the right stick to maneuver the claw around the logs and
to close the claw around them. You may have to open and close the claw a couple times to make sure they’re in there tight. Also make sure the claw is all the way up and tucked up against the truck. You don’t want it dragging on the ground.
Now just hold RT and head back the way you came and never let off the trigger. You cannot brake at all or the achievement will not pop. Luckily this truck doesn’t have a very high top speed. The trickiest part will be the right turn at the end of the paved street. Start your turn very early and you can cut it a little bit. Then just drive down the hill into the lumber mill area and as long as you never let off of RT and never hit the brakes you should get the achievement.
It may help to take a couple trips to learn the route before trying for the Unstoppable run. Here’s a screenshot of the path on the map. The K700 is just at the start of the uncharted area as you leave the lumber mill and head up the paved road.