For example, if you had an education startup that took, as payment, a contract from each student that they would pay an amount equal to 1% of their income tax to the education company each year. Would this be a legal arrangement, and would it be possible to arrange with an entity like the IRS to assist with these transactions?
Is it legal? Yes. You can make just about any payment arrangements you like.
You wouldn't be able to verify compliance, though, without some connection to the student's employer. The IRS won't tell you how much money a third party earned, or how much tax they paid, in a given year. Pretty sure they can't provide that info, which is why companies wanting to verify your income ask for copies of your tax returns rather than permission to get those returns.