In light of the fact that some incoming Muslim congresswomen may be sworn in on a Quran (by Mike Pence no less), it is worth remembering that the U.S. Constitution does not require an oath of office to be sworn on a Bible, or on any religious text for that matter.
Article VI, Clause 3, which covers oaths of office, states that while elected officials in both state and federal governments, as well executive and judicial officers throughout the country, are bound “by oath or affirmation” to support the Constitution, “no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States.”
Of course, we have already had Jewish, Hindu, Buddhist, and Muslim representatives and officials, both in Congress and throughout various other state and federal offices, swear on their respective religious texts (Fun fact: In 2007, Keith Ellison of Minnesota, the first Muslim congressperson, was sworn in on a Quran formerly owned and cherished by Thomas Jefferson.)
But many other officials, both religious and secular, have sworn on nonreligious texts, or nothing at all. John Quincy Adams and Franklin Pierce swore on a book of law; Lyndon B. Johnson was sworn in on a Roman Catholic missal; and Teddy Roosevelt, who had to take the oath in a hurry after William McKinley’s assassination, did without anything, since there was no Bible on hand.
Moreover, many Christians are forbidden by their teachings to swear on anything; both Herbert Hoover and Richard Nixon, who as Quakers could technically not swear on anything, could have “affirmed” rather than “sworn” during their oaths, though it appears they did not do so.