Not being Vader, I can’t say.
He may not have been entirely truthful.
I would say, though, that there may have been many potential options that could work. I haven’t read any of the Star Wars books that may have elaborated on what the authority structure was like, but just going by the three original films, I imagine it might go a bit like this:
The Emperor may have had succession orders that included Vader as potential heir to the imperial throne, and that didn’t have an exception for the Emperor’s mysterious or even murderous demise at the hands of Vader. That’d make it a simple matter of killing him.
If there were no succession orders favoring Vader, he may still have been able to assume authority as the new emperor, not just by virtue of his ominous reputation and position of authority as the Emperor’s hand, but also through Vader’s Force powers. Both Luke and Vader can be very persuasive using the Force, intimidation, and violence as(/if) needed.
The Empire’s command structure seems to have been very hierarchical and obedience-based, and the Emperor seems to have had various Dark Side staff and subordinates, who look to me like the sort of people who would do Vader’s bidding even if Vader had killed the Emperor, and I expect they were probably filling or pulling the strings of most of the higher positions of power, after the Senate was disbanded in Episode IV.
It’s possible there might have been some resistance from some factions withing the Empire, or even a great civil war, but Vader enjoyed crushing resistance, and considered himself more powerful than the Death Star. With no powerful force users to oppose them, Vader probably imagined the outcome was not much in question, or just didn’t care about risk because that’s the mindset of people caught up in the Dark Side.
