Does the vice-president serve at the discretion of the president, such that he or she can be removed from office at the president's will?
No. The President can not fire the Vice President.
Unlike the cabinet (e.g., Secretary of State, Attorney General, Secretary of Defense, etc.), and the White House Staff (e.g., Chief-of-Staff, Press Secretary, etc.) the office of the Vice President, his term and the requirements for his removal from office are all established explicitly in Article II of the Constitution.Office, Term—United States Constitution, Article II, Section 1
Removal—United States Constitution, Article II, Section 4
The executive Power shall be vested in a President of the United States of America. He shall hold his Office during the Term of four Years, and, together with the Vice President, chosen for the same Term, be elected, as follows [emphasis added]
The President, Vice President and all civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors.
Only Congress can forcibly remove a Vice President from office
The US Constitution only allows the Vice President to be forcibly removed from office via impeachment due to a high crime or misdemeanor (just like the President).
The President, Vice President and all civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other High crimes and Misdemeanors. (Article 2, Section 4)
Impeachment is solely the responsibility of Congress, so the President cannot unilaterally impeach the Vice President, or any other elected official in the federal government.
The House of Representatives shall chuse [sic] their Speaker and other Officers; and shall have the sole Power of Impeachment. (Article 1, Section 2)
The Senate shall have the sole Power to try all Impeachments. (Article 1, Section 3)
In other words, the House of Representatives has the power to start impeachment proceedings and the Senate has the power to perform the impeachment trial. If the Senate rules to convict the elected official, they are removed from office.
Note that the Vice President is also the President of the Senate (Article 1, Section 3). The Vice President presides in this role for any impeachment case (e.g. for a senator), except for that of the President because the constitution says that the Chief Justice presides in that trial.
Technically, that means that a Vice President undergoing impeachment would preside over their own trial. This might have happened with Spiro Agnew, Nixon's first Vice President, who was charged with extortion, tax fraud, bribery, and conspiracy. But rather than undergoing impeachment, he resigned in August 1973 (this was about a year before Nixon himself was involved with the Watergate scandal).
Also, the Vice President always has the option to voluntarily resign. This has happened twice: Spiro Agnew as described above, and John C. Calhoun in 1832. Calhoun was Vice President under John Quincy Adams and later Andrew Jackson in his first term. Jackson and Calhoun disagreed sharply on a number of matters, but as shown above, Jackson wouldn't have had the option to fire him if he wanted to. Instead, Jackson chose Martin Van Buren as his running mate for his second term (and the two were successfully elected). Calhoun resigned a few months before his term as Vice President was up to become a Senator for South Carolina.