I had to give up on trying to find out if a tattoo can count as consent on its own -- I would guess that it would be iffy territory unless you had it notarized and witnessed.

It might still be worthwhile to have a tattoo; it does tell them that you have given consent, meaning that they will make an extra effort to look for consent (In the US this means a state database). This would only be relevant if you are found without your drivers license/ID. There are a number of fringe cases where you might be found dead and dying without easy access to your ID, but they are admittedly rare. They are also more likely to cases where your organs aren't usable (fire, ravaged by bears, rip tide carries you out to sea). However, if the legal team gets any head start on finding a John Doe's organ donor status, on average this is likely to result in increased organ salvage.

Here's a revised suggestion, for social feasibility, effectiveness, and pain reduction: get a tattoo of a red heart and the words organ donor and your name in a protected area (e.g. on the side of your trunk, just below the arm pit). Until RDFI chips become common this is also probably one of your best protections against becoming a J. Doe (I mean, other than living a sane and safe life).