The rationale they had for storming the capitol was that the election was either rigged or unfair or fraudulent or in some other way not actually representative of the electorate. From their point of view, they have not done anything wrong in storming the capitol or stop the so called democratic process, they are ostensibly doing it in the name of democracy. Taking the VP's chair would from their point of view imply that they are just a bunch of thugs out to destroy the institution.Holy shit, the cognitive dissonance at 2:40 when some tacticool warrior tells another guy to get out of the VP's chair: "It is not our chair. We can't be disrespectful". From some guy who stormed the Capitol alongside people chanting treason and wanting to stop the democratic process. Tagging along with people wanting to lynch Nancy Pelosi that's cool, but some guy sitting in Pence's chair in the Senate? Nah man, that's crossing the line.
Granted, international observers saw no problem with the election itself, and the recounts did not give any indication that fraud was going on, so the storming was based on a false belief about the election, but that makes the insurrectionists mistaken, not cognitively dissonant.