Does a succeeding President select a new VP during succession?

Question

There is a clearly documented Order of Succession that is followed when the current President is incapacitated. This has been the subject of Films and TV Shows.

What normally happens in these situations, once the new President takes office? I believe, in short term situations, they'd just keep things moving until the President was able to resume his duties.

However, when the handover is permanent, does the new President then select someone to be his VP? Or do they just reappoint the lower positions and carry on without a VP?

If they do pick a new VP, given that the official VP is usually the Presidents running mate, are there any restrictions on who they are permitted to pick?

Answer 1

The 25th Amendment covers this situation. Before it was ratified, if the President died the VP office just stayed vacant until the next election. The amendment changes this to allow vacancies in the office of VP to be filled by the President with a twist: both houses of Congress (not just the Senate) have to confirm the new VP before they can assume office. This provision was used after Nixon's VP Spiro Agnew resigned over a corruption scandal. Nixon picked Gerald Ford as his new VP and Congress confirmed him. When Nixon resigned, Ford became the only President who never won a national election.

Answer 2

More relevant is that when Nixon resigned, Ford took over and chose Nelson Rockefeller to be VP (although Congress has to approve). Restriction is that the VP chosen has to be eligible to be President. (Older than 35, not having served more than a term and a half as president, natural born citizen, etc., not impeached and convicted and forbidden to hold office in the future,)