I work a lot with business applications and traditionally I've used the word "disapprove" as the counterpart to application approvals (e.g. disapprove leave application, disapprove waiver). But I've the sudden realization that in real life, the only times I've ever used the word "disapprove" is in the context of expressing negativity towards someone's actions and in more formal applications I would use the words "deny" or "reject" instead. Which is the best fit for business applications in this case? Is this a reasonable use for the word "disapprove"?
I would use either of the terms you identified, deny or reject. Deny is a more neutral phrasing, with the definition to refuse to agree or accede to. By not approving their requests, you are refusing to agree to them, i.e. denying them.
Reject has a similar meaning, to refuse to grant, or, to refuse to accept. I would argue, however, that reject has a more negative connotation than deny. I might be upset if my request were denied but I would be angry if it were rejected. Rejection has more of a sense of repudiation than denial. I don't know if this is a universal connotation; I'm sure the comments will discuss that.
I use decline for this. Your application for a waiver was declined. Your submission to our magazine was regretfully declined. And so on.
It carries the sense of "I think it is absolutely wonderful but alas I just can't take advantage of it on this occasion" and is certainly more positive than disapproved.
In formal business communications, I was advised to stick to Regret to indicate that a request/ application could not be approved. Yes, I too was surprised then, but in course of time, I discovered that it is well understood as a polite way of saying "No". There are other cultural differences across the world, and sensitivities, to be taken into account in drafting business communications.