Garage sales and lending your buddy five bucks are also part of that "informal" economy. Without cash, the government would be dipping its beak into those transactions as well.
I don't think it will actually work. Were the US to get rid of cash tax cheats and drug dealers would simply switch to GBPs, euros, or yen. Or drugs.
The only people who will be absolutely locked into the system are cubicle dwellers, but that's already pretty much the case.
Some guy who want some help with his yard-work one Saturday, probably is just going to skip hiring a day laborer if he has to register with the state as an employer and fill out all the local, state and federal income tax and employment tax forms for a days work. Even if the amount of money to pay that tax is trivial.
I think the informal economy exists to avoid both taxes and paperwork. And for some things the cost of the paperwork is significantly higher than the cost of the taxes.
http://www.investopedia.com/terms/d/double-irish-with-a-dutc...
To the extent that funds from the above-board economy transfer to the informal economy, the taxes are simply paid by the last person in the chain who paid taxes.
*they're narrowing now, but it is anyone's guess as to whether the big jump from the 2008 time frame is permanent or not.