Is the following grammatically correct?
Signing below acknowledges you agree with the terms.
Or does it need to be:
Signing below acknowledges that you agree with the terms.
I can't think of another transitive verb where this would be okay, but it doesn't have a definite "feeling" of wrongness, for whatever that's worth.
It looks like you have two questions: 1) is it transitive, and 2) is the reduction in sentence 1 correct. To both, the answer is yes, although with one caveat.
If you already have a good handle on what makes a verb transitive, SKIP THIS PARAGRAPH. A verb is transitive or intransitive based on its usage/context more than a quality that's innate to the verb. Thus, "I see a bear" is transitive (it has an object), while "I see" is intransitive. There could be an argument about how these two versions of "see" have different meanings, but this argument can be answered with another example: Are you hungry, little Timmy? "No, I ate (a sandwich)." The semantic denotation remains consistent in that example. Beyond this, remember that complements AND objects can make a verb transitive. The verb acts upon something else (and it isn't a copular verb, like "to be").
In your sentence, we have a transitive verb leading to a verb complement that takes the form of a noun phrase. The complement can be replaced by "something," which is a common test for "noun-ness" used by grammarians. "Signing below acknowledges SOMETHING." The "that" in your sentence is a "complementizer." We can reduce relative clauses that feature "that," but can we do the same when "that" sets off a complement?
"I heard you changed jobs" is in acceptable use, but "I love you're so funny" is not. Still, it's easy to find more examples of a transitive verb linking with a verb complement: "I see (that) you're angry," "I wish (that) you would visit more often," or "I understand (that) you're angry." The last is quite similar to your sentence.
I don't think we can say that your construction is categorically correct (i.e. we did find a transitive verb that needed the complementizer "that"), but I think that we can make a case that common usage of the verb "acknowledge" dictates "Signing below acknowledges you..." and "S B A that..." are semantically the same.
As it sounds like a legal document, there may be room for hairsplitting, but there's nothing incorrect with either of those sentences when it comes to everyday use. If you'd like input on the legal ramifications of the reduction, I invite someone else to join in the fun (I heard of a case where a comma voided a million-dollar contract in Canada, and after I read the excerpt, couldn't tell you what the difference was!)
