I've found some discussion on what happened to the office of Pontifex Maximus during the Roman Empire, but none on what happened to the Flamen Dialis. Did this office die out fairly early in the empire? It appears it already was vacant for nearly 70 years during the first century BCE.
Despite the prestige, I imagined that the many taboos associated with the office must have made it rather inconvenient to be Flamen Dialis.
You are right about the vacancy. The flamen dialis L. Cornelius Merula had been in that role and committed suicide when Marius took over the city. In 87 or early in 86 BCE, the young Julius Caesar was slated to take over the position by the Marian government, but before that was enacted, Sulla retook the capital and invalidated all the Marian actions. So Caesar went on his merry way and the vacancy just seems to have sat there unfilled. It was an onerous office, but I don't think that's a good explanation -- there were plenty of minor Roman aristocrats who would never attain the heights of political achievement seen in others, and they would certainly love to have such an ancient, awe-filled, and prestigious office. The flamen dialis had many inconveniences: he could not be away from his bed more than three consecutive nights (and thus could never govern a province or lead an army); he couldn't touch or even look at a horse; he couldn't have any gemstones on his rings; couldn't swear any oaths; couldn't go out without his weird headgear; he could not touch a knot or have on in or on his clothes; he was forbidden either to touch or to name a dog, a she-goat, ivy, beans, or raw flesh (yep); et cetera!
But there were also a lot of perks: he got a lictor to guard him; he got to wear the toga praetexta in public; he got a curule chair and could sit in the Senate; he got the 2nd-best seat at any feast; etc.
I think the Occam's razor answer is that the 'succession' was disrupted after Sulla and there just wasn't very good machinery to get it sorted out afterwards. During the joint consulships of Crassus and Pompey in 70 BCE, they held a ritual lustrum (purging/cleansing of the city) and the vacancy should have been rectified then, if ever, but it just wasn't. The duties of the office seem to have been carried out by other pontifices without much issue. It remained unfilled until 11 BCE, when Augustus finally appointed Servius Malugenensis. I can find no further references to the office, so I think your guess might be right that it just petered out. If so, I find that very strange. The Romans were very big on ancient custom and were incredibly conservative about such things. There must be an explanation but I can't find it.