Correct Idiom: "Require X to do Y" so Discard D and E
In B "them" is wrong
In A either you can use "Prohibit X from Y" or "Prohibit X". "Prohibit That" is always wrong

Se left out with C.

In an attempt to guarantee the security of its innovative water purification method, the company required each employee to sign a confidentiality agreement prohibiting that its water purification methods be disclosed to companies using an analogous purification process.

A. prohibiting that its water purification methods be disclosed to companies
B. prohibiting them from the disclosing of its water purification methods to any company
C. prohibiting disclosure of its water purification methods to any company
D. that would prohibit them from disclosure of its water purification methods to companies
E. that would prohibit its water purification methods to be disclosed to a company

The issues:

1. The following idioms are correct:

==> "Prohibit X from doing Y".
==> Prohibit X, where X is a NOUN.

2. The following usages are incorrect:

==> Prohibit that X be Y.
==> Prohibit X to be Y.

Now, come to the sentence.

On the basis of the idioms, we can eliminate the options A and E. The pronoun "they" does not have a clear reference in the B and D.

Answer is C.