Suppose Boehner resigns, leaving the US House of Representatives without a Speaker. What happens next? Is there some process for operating the House without a Speaker? Or is the election of a new Speaker the required (by law or internal rule) next item of business? If there is such a requirement, what are the details of it?
The election of a new speaker is the first order of business of a new sitting of the house, or on the death or resignation of the existing speaker - so yes, the house will be hamstrung until a new speaker is elected.
As a historical example, in 1854 it took 9 weeks and over 130 votes before a new Speaker was elected - prompting a change in rules to make winning easier (The winner only has to receive the most votes regardless of the number of ballots cast rather than an absolute majority of > 50% of the congressional seats). This time around, they don't have the luxury of that much time as they MUST pass a spending bill by Dec 11th or a government shutdown will occur.