I can't see any way to do so through the constitution, but I am consistently surprised by the kinds of laws that the federal government can make, so I'm obviously not well-versed in the possibilities.
Yes. The federal government has the authority to mandate this if it wished to do so.
Not clear if it would be a 1791 constitution privileges and immunities clause enforcement, a 14th Amendment equal protection and privileges and immunities enforcement, or an interstate commerce enforcement, but it could clearly do so.
It is a straightforward and not-uncommon position that the Privileges and Immunities Clause in Article IV of the U.S. Constitution already prohibits (public) universities from charging out-of-state students higher tuition. The clause reads as follows:
The Citizens of each State shall be entitled to all Privileges and Immunities of Citizens in the several States.
As examples of law review articles arguing or discussing this point, see "Privileged for Being Stationary: Why the Practice of Differentiating Between In-State and Out-Of-State Tuition Rates are Unconstitutional" (or this link) and "On the Constitutionality of Charging In-state Tuition" and "The Right to Residency: Mobility, Tuition, and Public Higher Education Access".
Note that there is also the Privileges or Immunities Clause of the Fourteenth Amendment, as well as a presumed right to travel, that may cover this situation. These are discussed in the cited articles above.