No, votes would not have to be postponed.
The Constitution has nothing specific to say about this. (It's not that long - you can and should read it through and check for yourself, and searching is even easier.) The Constitution's only reference to the Speaker of the House is Article I Section 2: "The House of Representatives shall chuse their Speaker and other Officers [...]". (Other than the 25th Amendment which prescribes the role of the Speaker in receiving declarations as to whether the President is incapacitated.)
The House's procedures, and the Speaker's role in them, are left up to the Rules of the House of Representatives, which the House makes for itself. (US Constitution, Article I, Section 5: "Each House may determine the Rules of its Proceedings [...]".) Rule I, Section 8 provides:
- (a) The Speaker may appoint a
Member to perform the duties of the
Chair. Except as specified in paragraph
(b), such an appointment may not extend beyond three legislative days.
(b)(1) In the case of illness, the
Speaker may appoint a Member to perform the duties of the Chair for a period not exceeding 10 days, subject to
the approval of the House. If the
Speaker is absent and has omitted to
make such an appointment, then the
House shall elect a Speaker pro tempore to act during the absence of the
So if the Speaker is ill, she can appoint a temporary substitute (Speaker pro tempore), who can preside over all House business, including votes. If she cannot or does not do so, the House may elect a Speaker pro tempore with the same authority. (That election itself would be presided over by the Clerk of the House, an administrative official, as specified by Rule II Section 2(a).)
Either way, there would be no need for votes to be postponed.