A certain school's admissions policy requires of both undergraduate an

globaldesi , ah, now I understand.
To pose questions that are clear, stress the word or phrase that bothers you, as you did the second time around:
It is the presence of "of" that bothers me.

Your other question is ambiguous: Will require of both not need "to" rather than "that"?
Are you asking about the phrase "require of both," or
are you asking about the word "of" in the phrase "require OF both"?

I edited my previous response.

Overall: REQUIRE OF must have a that, not a to.
Require that must be in the command subjunctive form discussed above.

Two more thoughts —

First, I think it is easier to eliminate on the basis of the construction of "require to" rather than "require that."
The construction of "require to" is incorrect, so options C, D, and E are eliminated.

Second, the idiomatic constructions for "require to" and "require that" are not hard to remember (see below).
The verb "require" is a ripe target for testing.

We have two possible constructions:
-- The policy requires someone to do something. Require + object + infinitive
-- The policy requires that someone do something. Require + that clause in subjunctive command form

Require + infinitive is structured incorrectly in this question.

If the policy requires someone to do something the structure is
require + VERB object (a something or someone) + infinitive

I require a doctor to treat my injured arm.

"doctor" is the object of the VERB. Whom do I require? The doctor.

The verb must have its own object, someone who (or something that) receives the command (is the target of "require").

But prepositions also need an object.
Correct: The child was afraid OF monsters in the closet.
Wrong: The child was afraid OF in the closet.

One word cannot simultaneously serve as the object of a preposition AND the object of a verb.

I require him to write well.
I require Tom to write well.
Whom do I require to write well? Tom. Him.
The verb has its object.

I require OF Tom to write well?
Now we have a verb, require, that needs an object, AND a preposition, OF, that also needs an object.
My verb's object just went away. Tom became the object of the preposition OF.
Take out the prepositional phrase just to see what happens.
I require [] to write well. Nonsense.

The object of the preposition "of" cannot also be the object of the verb "to require."

The policy requires OF the schools to practice gender equity? NO. Wrong.
"Schools" is the object of the preposition "of."
"Schools" can be the object of only one thing: the verb OR the preposition, but not both.

Remove the prepositional phrase in order to see what happens.

Now we have, "The policy requires [] to practice gender equity." That sentence is nonsensical.

Eliminate C, D, and E

Between A and B: "should" is never part of command subjunctive (despite common usage in British English)
Eliminate B.

Idiomatic construction

Answer A

Short version, all correct, and if not on this list, not correct
Require X to (do) Y
Require that X do Y
Require OF X that it do Y

When we use the word "of" after require, we cannot use the infinitive construction.
Rather, we must use the command subjunctive construction.

If I state a command or directive to an audience, after all, I require OF the audience that it do something.

This question and idiomatic construction
CORRECT
• Require someone TO do something: The policy requires the two schools to practice gender equity.
• Require THAT someone do something: The policy requires that the two schools practice gender equity.
• Require OF someone that she or he do something: The policy requires OF the two schools that they practice gender equity.

INCORRECT
• Require OF someone to do something
• Require OF someone that she or he SHOULD do something
(should is implied in a command)

Your question, reformatted: Will require "OF" [both undergraduate and graduate schools] need "to" rather than "that"?

Answer: No. In fact, quite the reverse: the OF in require of both
demands the word that, not the word to.

For one thing, if we use require to, we have no direct object; the phrase "of both schools" is not an object of the verb require.

For another, using require to in this situation is not idiomatic.

Hope that analysis helps.