VP can resign at any time though. I'd have to double check, but I don't think there is a line if succession for the VP, since the President is over the VP, they can just select a new VP, although Congress may have to have a majority vote to accept the appointment.
Now to go check how good my HS government class was almost 20 years ago...
Edit : 25th Amendment, Section 2
Whenever there is a vacancy in the office of the Vice President, the President shall nominate a Vice President who shall take office upon confirmation by a majority vote of both Houses of Congress.
I actually can't find anything (quickly) about when/how the VP can resign, but I assume they would have to formally submit their resignation in writing, or verbally and directly to the President, who would accept the resignation. As far as I can tell, the VP could resign right after being sworn in.