Easy way to Catch the Tsuchinoko is: Make sure you are in the Area where you fight The Fear at, Kill all living creatures, set up 3 mousetraps around the entrance of the Warehouse (that way you can clear see your mousetraps), go into the Warehouse and come back. All 3 Traps are filled and one will at least be the Tsuchinoko. Crawl over your mousetraps to pick them back up and you are good to go.
The Prison break (important): After you've been captured, you'll lose all your gear, and that of course includes your Tsuchinoko. Just proceed normally to the waterfall with Eva. After you've met up with Eva in the cave, and the cutscenes are done, go out the way you came in, so you're back outside again. The Tsuchinoko will be on the river banks, crawling along. This time, it will be visible. It should be on the river bank to your left. Just look around for it there, you should still be in sight of the waterfall, it's close to it. It might be worth looking at it in the survival viewer before you lose it, so you know what to look for. It does stand out a bit, with it's way of movement.
If you can't see it to start, just keep looking. Just shoot it with your MK22 when you find it. Don't kill it, as you will need it alive. Pick it up again, and you're set for the rest of the game, just make sure you don't eat it or dispose of it.
If you keep it alive until the end of the game, where Ocelot kindly chucks it off the plane with your items, you will gain infinity face paint and the rank of Tsuchinoko. This face paint will give you unlimited ammo when worn.
Just remember all I've said, and you should be able to catch and get all the rewards from your Tsuchinoko.