Is it a requirement of holding congressional office that one must swear an oath to uphold the Constitution of the United States?
The law regarding, and the wording of, the oath is given in 5 U.S. Code § 3331. It is also not limited to just Congress:
An individual, except the President, elected or appointed to an office of honor or profit in the civil service or uniformed services, shall take the following oath: “I, AB, do solemnly swear (or affirm) that I will support and defend the Constitution of the United States against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I take this obligation freely, without any mental reservation or purpose of evasion; and that I will well and faithfully discharge the duties of the office on which I am about to enter. So help me God.” This section does not affect other oaths required by law.
As an aside, the President is exempted in the law above, because that oath is specifically prescribed in Act II, Section 1 of the Constitution:
Before he enter on the execution of his office, he shall take the following oath or affirmation:--"I do solemnly swear (or affirm) that I will faithfully execute the office of President of the United States, and will to the best of my ability, preserve, protect and defend the Constitution of the United States."