The whole point of the "envelope system" as I understand it is that it makes it easy to see that you are staying within your budget: If the envelope still has cash in it, then you still have money to spend on that budget category.
If you did this with a bunch of debit cards, you would have to have a way to quickly and easily see the balance on that card for it to work. There is no physical envelope to look in. If your bank lets you check your balance with a cell-phone app I guess that would work. But at that point, why do you need separate debit cards? Just create a spreadsheet and update the numbers as you spend. The balance the bank shows is always going to be a little bit behind, because it takes time for transactions to make it through the system. I've seen on my credit cards that sometimes transactions show up the same day, but other times they can take several days or even a week or more. So keeping a spreadsheet would be more accurate, or at least, more timely.
But all that said, I can check my bank balance and my credit card balances on web sites. I've never had a desire to check from a cell phone but at least some banks have such apps -- my daughter tells me she regularly checks her credit card balance from her cell phone. So I don't see why you couldn't do it with off-the-shelf technology.
Side not, not really related to your question: I don't really see the point of the envelope system. Personally, I keep my checkbook electronically, using a little accounting app that I wrote myself so it's customized to my needs. I enter fixed bills, like insurance premiums and the mortgage payment, about a month in advance, so I can see that that money is already spoken for and just when it is going out. Besides that, what's the advantage of saying that you allot, say, $50 per month for clothes and $100 for gas for the car and $60 for snacks, and if you use up all your gas money this month than you can't drive anywhere even though you have money left in the clothes and snack envelopes? I mean, it makes good sense to say, "The mortgage payment is due next week so I can't spend that money on entertainment, I have to keep it to pay the mortgage." But I don't see the point in saying, "I can't buy new shoes because the shoe envelope is empty. I've accumulated $5000 in the shampoo account since I went bald and don't use shampoo any more, but that money is off limits for shoes because it's allocated to shampoo."