...OR in other words, is there a scenario where the President gets forced out in a way that prevents the Vice President from succeeding him or her?
The easy answer: Impeach and remove the VP first, and then impeach and remove the President before confirming the President's new pick for VP.
The more speculative answer: Although it has never happened, it seems plausible (in my non-expert opinion) that the US House of Representatives could approve multiple articles of impeachment for different individuals (like the President and Vice President) simultaneously (or immediately after each other), and pass them all to the Senate at the same time. Whether the Senate could try them simultaneously is a more difficult question, see below.
The crux of my argument is that impeachment is already handled as a collection of "charges" (the articles of impeachment). President Nixon faced 3 articles that were never voted on; President Bill Clinton faced 2 articles for which he was acquitted by the Senate; Judge Thomas Porteous Jr. faced 4 articles for which he was convicted and removed.
I couldn't find any constitutional or procedural prohibition for the House considering impeachment of two people as part of the same proceedings, and if the committee is investigating the same set of facts, it makes a lot of sense to not have to repeat their work.
In this hypothetical, the President and Vice President would still each get their own personalized set of Articles, recommended by the Judiciary Committee, and voted on by the entire US House. See this article for a general outline of the process, or this Congressional Research Service Report on Impeachment and Removal (PDF) for more detail:
If the full committee, by majority vote, determines that grounds for impeachment exist, a resolution impeaching the individual in question and setting forth specific allegations of misconduct, in one or more articles of impeachment, will be reported to the full House.
What is less certain is whether the Senate trial could occur for both defendants together. The Senate's "Rules of Procedure and Practice in the Senate When Sitting on Impeachment Trials" (PDF) seems written with a single "respondent" in mind. For example:
I. Whensoever the Senate shall receive notice from the House of Representatives that managers are appointed on their part to conduct an impeachment against any person and are directed to carry articles of impeachment to the Senate...
XVII. Witnesses shall be examined by one person on behalf of the party producing them, and then cross-examined by one person on the other side.
XXII. The case, on each side, shall be opened by one person. The final argument on the merits may be made by two persons on each side.
If there are two defendants/respondents, then there are more than two "sides".
But it's clearly in the power of the Senate to revise these guidelines, so that may not be an impediment.
And again, if the same set of facts is at issue (requiring the same witnesses and same testimony), then it may in fact be worth the Senate's effort to revise their procedural rules to make this possible.
In any case, at the conclusion of the trial each article is voted upon separately by the Senate:
XXIII. An article of impeachment shall not be divisible for the purpose of voting thereon at any time during the trial. Once voting has commenced on an article of impeachment, voting shall be continued until voting has been completed on all articles of impeachment unless the Senate adjourns for a period not to exceed one day or adjourns sine die. On the final question whether the impeachment is sustained, the yeas and nays shall be taken on each article of impeachment separately; and if the impeachment shall not, upon any of the articles presented, be sustained by the votes of two-thirds of the Members present, a judgment of acquittal shall be entered; but if the person impeached shall be convicted upon any such article by the votes of two-thirds of the Members present, the Senate shall proceed to the consideration of such other matters as may be determined to be appropriate prior to pronouncing judgment. Upon pronouncing judgement, a certified copy of such judgment shall be deposited in the office of the Secretary of State. A motion to reconsider the vote by which any article of impeachment is sustained or rejected shall not be in order.
For more fun reading, see this even lengthier Senate PROCEDURE AND GUIDELINES FOR IMPEACHMENT TRIALS, which includes details of past impeachment proceedings.
- Whenever there is a vacancy in the office of the Vice President, the President shall nominate a Vice President who shall take office upon confirmation by a majority vote of both Houses of Congress.
So if congress removes the vice president it can fail to confirm a new one before removing the president.
- The administration does something bad and congress finds out.
- Congress (house) begins impeachment proceedings on the vice president.
- Congress (house) successfully impeaches the vice president.
- Congress (senate) tries the vice president.
- Congress (senate) convicts and removes the vice president.
- The administration nominates a new vice president.
- Congress (either) does not confirm the new vice president.
- Congress (house) begins impeachment proceedings on the president.
- Congress (house) successfully impeaches the president.
- Congress (senate) tries the president.
- Congress (senate) convicts and removes the president.
- The Speaker of the house becomes the chief executive.
There is some flexibility in some timings; 1-5 must come before 6&7 and 12 can only come at the end, but 2-5 are independent of 8-11. 11 must come after 5 as a requirement of the question.