Hypernym for "approve" and "reject"

I would prefer to see Approval Requests as the button. The request is for approval, so the admin can either accept or deny the approval.

Judgment could work in general, but probably not in your specific context.

TL;DR - "Approval"

If the act of approving or rejecting the request is simply the click of a button, then obviously there could be two buttons labeled "Approve" and "Reject" next to the item, but this is clearly a design consideration and nothing to do with the use of English.

If the design is to take the approver (there is a hint already) to a separate screen where the act of approving or rejecting the item must be carried out in more detail, then I would also go with "Approval" as the button label to represent the act of approving (which may still result in rejection).

You generally say that something is sent for "approval" and not "rejection". However, that does not mean it will definitely be approved. It can still be rejected. Similarly, the person doing the approval can be termed an approver, but is never termed a rejecter.

Therefore I maintain that "Approval" as a noun is a good collective term describing the act of approval, which may result in something either being approved or rejected.

From working with enterprise systems, where this query originates, my experience is that the term "approval", as a matter of common usage, is generally deemed and understood to be such an act which may result in either approval or rejection.

Handle is another one that immediately springs to mind. When you approve/reject requests, you handle them. In fact, handle requests is a very common collocation.

Use comment on request. It's applicable in context.

How about 'decision'? As a decision can be either an approval or a denial but never 'pending'.