If Congress wants to hold someone in contempt for failure to comply with a subpoena, they can do several things:
(1) Hold congressional contempt proceedings. These are quasi-judicial proceedings, rooted in the constitutional investigatory and legislative power of Congress, in which the members of Congress themselves act as judges, juries, and prosecutors. It is very important to note that this is not a judicial process. The chamber of Congress in question has the power to direct the Sergeant at Arms to arrest someone, bring them before the chamber, and put them in jail, all without seeing the inside of a courtroom.
See McGrain v. Daugherty, for an example of this upheld by the Supreme Court. The Court held that such powers were “necessary and proper” for Congress to carry out its legislative function.
This is the “historical” method by which a chamber of Congress has enforced its subpoenas. It was employed some dozen or more times up until 1934 but it was deemed too time consuming because it required the attention of the whole chamber, sometimes for more than a week. In the 1850s an alternative procedure was hotly debated and eventually created. But this method still exists. And this is the source of what is referred to as the “inherent contempt” proceeding.
For completness’ sake I should mention that individuals imprisoned under this procedure may petition for habeas relief from federal courts. They therefore will have some opportunity (though likely limited) to raise defenses (for example, Fifth Amendment or executive privilege) and challenge the validity of the contempt finding. On the other hand, there is general consensus that this type of imprisonment is not a criminal penalty and therefore is not subject to the presidential pardon power.
(2) Certify a contempt citation and deliver it to the U.S. Attorney for the District of Columbia. This is a multi-step process. First, the committee (or subcommittee) that is seeking the citation must approve it by majority vote. Then, it gets handed up to be read and voted on by the next higher authority (either the full chamber or the committee). Once the full chamber has certified the contempt citation, a report is prepared and presented to the U.S. Attorney. The U.S.A. then has a duty to present the contempt citation and report to a grand jury in order to get an indictment for contempt of Congress. See 2 U.S.C. § 194.
This is the modern, statutory method that Congress has used to lay contempt charges on reluctant witnesses. This is criminal contempt. A contempt indictment then goes through the normal judicial process, meaning the subjects of the citation will have the opportunity to raise defenses or outright challenge the validity of the contempt citation. Individuals punished by criminal contempt may seek a presidential pardon.
(3) The Senate has another option. Under a newer law, deemed necessary during President Nixon’s battles with Congress, the Senate can seek a civil contempt order from the U.S. District Court for the District of Columbia. Defiance of this order would be contempt of court, rather than contempt of Congress. The district court will examine the validity of the civil contempt citation. Individuals facing a civil contempt order will have the opportunity to challenge it and raise defenses, just like in the criminal contempt proceeding. This civil contempt option is sharply limited. It may not be employed against federal employees acting in their official capacities.
(4) Finally, either chamber could file suit in civil court for declaratory and injunctive relief after it has issued a subpoena that has been ignored. This is not a contempt proceeding, but it would be one way of enforcing subpoenas if any of the contempt procedures above failed. This is the least tested option, especially when the subpoenas are directed at the Executive Branch, because it raises so many separation of powers questions. Congress would be asking the Judiciary to force action by the Executive. Courts are usually very unhappy about doing that and usually refuse to rule on such cases, citing the political question doctrine. The doctrine provides that some questions are more properly committed to a coequal branch of government, that the court has no standards with which to resolve the dispute, or that it would be imprudent to interfere. This procedure also allows the subpoenaed individuals to present defenses and challenges to the validity of the Congressional subpoena.
In short, Congress has several options for enforcing subpoenas, but in each case the subpoenaed individual will have the opportunity to raise defenses and challenge Congress’ contempt order. Given recent events, of most interest for us will be claims of executive privilege.