Thanks for climbing in (or is it on) to what could become a lively and hopefully a interesting debate.
Yours is a fair question, and I'm now going to be rude - my late mum always said I should never answer a question with a question - BUT:
First, I'm assuming we are talking about the US banking system? Yes, No? So: How can businesses do business in the US of A without having a US Bank account and how can US customers do business with any businesses, domestic or foreign, without having bank accounts in the US of A? And yes, I have heard of internet banking. But have you tried as a US citizen, resident in the US, to open a bank account in, say, the UK? Not very easy and not very cost effective to run, I would imagine. I suppose businesses could refrain from doing business in the US of A, true, and cut out business in possibly the biggest economy in the world (don't know for how long, though). I don't think they would do that. Perhaps I am being naive but would you seriously place your life's saving and monthly income under your mattress and transact only with cash? To avoid a 1% levy that replaces all other forms of state revenue? I know people are not very rational, economists learn that over and over again, but surely it's better than an insurance policy, and probably cheaper (what would you pay to insure money under a mattress?) and more effective (how do you pay your bills with an insurance policy?).
If businesses think they can outsource their banking services - well, TEAL would have a regulatory framework, I imagine, which would define things like who would be deemed to be acting as a bank and be required to collect teal, to how salaries and wages can be paid and how proceeds from business transactions can be received (eg into a domestic bank account on both scores) and what happens to folk who deliberately undermine and subvert the tax system, which probably wouldn't be very pretty (pretty much all these regulations probably already exist). And all that just to avoid a 1% levy that is going to make your dysfunctional economy work better and be good for business - toss vat, payroll taxes, income taxes, profits taxes, fuel levies and whatever other garbage your own resident friendly neighbourhood governments dump on you, and all the administrative claptrap that goes with it?
Perhaps I'm just being stupid, but the 1% levy certainly appeals a great deal more to me than the near 60% of income it is estimated every salary and wage earner is paying in this country. Perhaps the tax laws in the USofA are less stringent than in SA, but I seriously doubt that. But there are also up-sides for businesses. Do you think they administer all the taxes they have to collect and pay over without serious cost and inconvenience? It is almost a win win situation.
Of course, there are those who previously have been able to avoid taxes (they start losing, relatively speaking) - NGOs, Churches, organized crime (in whatever guise their activities take) and folk who transact asset purchases with cash that has never been declared, folk with clever tax lawyers and accountants and then there are the investors and the day traders and high speed traders, those who game the system and make huge profits at tiny margins, they will be hit, doubtless many forms of trading will become unprofitable, but would that be a bad thing? And doubtless there are many others I am not even aware of - they will all be drawn into the net - unless they can find a way of existing without using a domestic bank account - and as I hinted earlier in this post, that would probably be illegal. And also of course, there will be those who still avoid tax, or teal, but they would be an endangered species, I think.
I'm not sure I have answered your question very well. Let me know. I'm happy to try again.
To answer your first query more directly: Economic activity is measured by the aggregate of all the payments and receipts in all a nation's bank accounts. At a micro level, each bank account can be levied on all the receipts and payments made into and from the account. You are not levying the banks, you are levying their clients. You do however levy the banks inasmuch as they use their own banking services to transact their own business. There they are their own clients and as such their transactions are part of a nation's economic activity.
In South Africa, we have a handy central point, being the National Payments System or NPS, which 'clears' the transactions between banks. We have a fair idea of the volumes of money that flow through the system on a daily basis, and the volume is huge. It is that volume that we wish to tap, but at a micro level within each individual bank account. There are of course transactions that occur within banks, between clients of the same bank, that do not pass through the NPS, but we have a fair idea of what that volume represents as a percentage of the NPS volumes. So that is how we are able to measure the economic activity of our nation.
There must be similar institutions in other countries, although they are not necessary for the implementation of TEAL, which I repeat, is levied at individual account levels, but they are neat places at which to measure total economic activity and to do your sums with.