According to 2 USC 25: Oath of Speaker, Members, and Delegates. (Text contains those laws in effect on January 4, 2023:)
“At the first session of Congress after every general election of Representatives, the oath of office shall be administered by any Member of the House of Representatives to the Speaker; and by the Speaker to all the Members and Delegates present, and to the Clerk, previous to entering on any other business; and to the Members and Delegates who afterward appear, previous to their taking their seats.”
Essentially, most interpretations of this section state that the oath of office must be administered by the House Speaker. (Can't Members swear each other in?) Until then, all representatives are just Representative-Elects, and don’t hold any actual power. Without a Speaker, there are only 100 members of Congress (the US Senate), and an interim speaker cannot administer the oath.
It appears that "members are operating under the rules for the last Congress" but the legitimacy of this approach is uncertain. Members and staff are uncertain if they will get paid or if they are covered by health insurance. Since the Senate is the US Congress at this point, the House might have be paid by the Senate and go under the Senate's health plan until this situation is settled.
(Research Assistant: Matthew Weinstock, American University)