It's not a right, it's a power granted to congress. The constitution doesn't guarantee postal service to anyone.
Article I, Section 8, Clause 7 grants congress the power to establish a post office and postal roads. It's interpreted as being implied that it grants this post office the right to carry and deliver mail, as well as define where those offices are and the routes they must follow. This doesn't require congress to do it, it merely grants them that power. Congress could essentially end their use of that power at any time, as there's nothing preventing them from doing so.
They could just as easily establish a postal service and define those services for a private company - which is essentially what the post office is now. It's a private company that isn't allowed to define it's own practices, they have to petition congress for them. Or, they could simply say "Anyone that wants to carry mail can do so, WDGAF anymore."