If the USPS went out of business, that would mean UPS would be in big trouble because they ironically need the USPS to finish most of their routes; they don't have enough locations to serve everywhere, so rural areas would be wiped off and big cities would thrive.
Amazon despite its size, would be unable to meet the demand and become a private carrier. It would probably threaten them in a good way.
There would be no way a small business would survive, and would kill off some medium sized business. So with them out of a job, you would be looking at a depression event since no one would be buying houses, or cars. Which in turn eliminates jobs with construction and auto repair since they wouldn't be needed.
The only good thing about this, is that it's in the constitution itself.
Article I, Section 8, Clause 7 of the United States Constitution, known as the Postal Clause or the Postal Power, empowers Congress "To establish Post Offices and Post Roads".
And in order to change it.
Under Article V of the Constitution, there are two ways to propose and ratify amendments to the Constitution. To propose amendments, two-thirds of both houses of Congress can vote to propose an amendment, or two-thirds of the state legislatures can ask Congress to call a national convention to propose amendments. To ratify amendments, three-fourths of the state legislatures must approve them, or ratifying conventions in three-fourths of the states must approve them.