Secretary of the Army

Got this off the USNA website because it has a numerical breakdown of slots.
Who can nominate?
The first three bullets = the MOC Slots

The Sect. of the Navy nominates the military (service-connected admissions) so 170 slots for Navy & Marines; I assume same for Army

All Navy ROTC units and all Navy and Marine Corps JROTC units (& Honor Units) are eligible to nominate three candidates each, again assume same for Army

FYI: Army JROTC is in 1645 schools; don’t know if every school is an individual unit.

Assume the last two bulleted categories fall the Sec. of Army as well.

Looks like Title 10 of the US Code would have any additional if needed.
For USNA, SecNav is a nominating source but one to which candidates can't apply. SecNav noms are used for NAPS/Foundation candidates who don't obtain noms (or aren't slotted to) other sources. Candidates who are appointed from the national pool are also typically lotted to SecNav.

Don't know about USMA's process specifically but presume it's at least somewhat similar to USNA's.
Specifically for USMA, they are lumped together as "Service Connected" nominations and the eligibility and categories are listed in the cadets portal. Going from memory, some require the candidate to send in all of the information, dependent of career service members requires parents service records, and others require additional paperwork, commander's endorsement for active duty soldiers or PMI recommendation for ROTC components.