Presidential noms are handled by the respective service academy, and really don't have much to do with the actual office of the President, whoever it might be. Those noms are essentially non-competitive, in the sense that the candidate just has to satisfy the eligibility requirements. Getting that nom doesn't equal appointment, as that is a competitive process with others in various nom categories. The SAs who participate in the nom process are trusted to assign those noms to eligible candidates. Most candidates apply for as many categories of noms as they are eligible for. SA websites will have that info.
VP noms, only 5 nation-wide, are nationally competitive, and are handled by the Office of the Vice President. There will be long-time staff members who have been handling the rack-and-stack on these packages for years, perhaps in conjunction with a review panel. They will develop a list of the top recommendations, and it will move through the VP's office in the document signature queue as routine business, regardless of who is in the chair. It may even be signed by a staffer with delegated authority, using an auto-pen or e-signature.