Qfyilyi

In the current Congressional elections, the control of the House of Representatives is changing from one party to another, and the current Speaker did not run for re-election.

In this situation, is there a period of time where the post of Speaker is vacant, creating a gap in the Presidential line of succession? Starting and ending when?

There will be no gap in succession. The House will elect a new speaker and Ryan will tender his resignation on the same date. The date of the election of the new speaker will be when the House decides to vote on the new speaker during the last week of the current lame duck session, which ends January 3, 2019.

An election for the Speaker of the U.S. House of Representatives will
take place during the first week in January 2019, during the opening
day of the 116th United States Congress, two months after the 2018
elections. 2019 Speaker of the United States House of Representatives election - Wikipedia

The Democrats have a majority in the new Congress, so presumably the new speaker will be a member of that party.

There will be a gap in succession, but only briefly.

Speaker Paul Ryan's term extends past the election through the end of the 115th Congress, to 11:59:59 p.m. on January 2, 2018.

From there, the speakership is vacant until the 116th House of Representatives elects a successor. Traditionally, the House convenes at noon on January 3 of the year after an election. The election of the new speaker is -- as far as I can remember -- always the first vote taken in the new term.

The vote usually comes after a roll call, a prayer, and the Pledge of Allegiance, so you can usually expect the speakership to be filled by 1 p.m., leaving a gap in the office of 12 to 13 hours.