I think that The Initiative had to be fairly new when S4 begins. They're completely useless and spend most of their time running around like a chicken with its head cut off.
Granted, the Initiative is the epitome of ineptitude, poor leadership, and ill defined mission, but common logic would have to indicate that they simply couldn't have just suddenly popped up, fully constructed and manned,over the summer between Seasons 3 and 4. Their underground headquarters alone would have taken at least a year of intense labor to construct, and that construction would have had to be carried out in almost total secrecy.
The Initiative might very well have been in Sunnydale, at least in the construction stages, even before Buffy got there, possibly since the college was first constructed or even earlier. We know that the Initiative was founded sometime neat the beginning of World War 2, and the demonic activities in Sunnydale would have been exactly the sort of thing that they would have been investigating.( Remember them recruiting Angel to rescue the American submarine crew?That might have been the inspiration for the ultimate development of the chip implanted in Spike's brain. If they could prevent a vampire from killing people,the next step would have undoubtedly been a way to control their activities,creating an army of literally super soldiers. Just keep them out of the direct sunlight. Bullets wouldn't stop them, and by the time the enemy realized exactly what they were up against, half or their own troops would have been turned, chipped, and fighting against their former commanders.)
Just because we never saw them, doesn't mean that they weren't there. The Initiative didn't concern themselves with apocalypses, they probably didn't believe in them at all, and they wouldn't have even noticed the activities of the Master in season 1 or Angelus in Season 2 as anything more than a temporary increase in HST activities
The Ascension would have been impossible to ignore, but covering up something like that, and there was undeniably a major cover up required to explain all of that, convincing the news media that the whole thing was the result of LSD slipped into the pre-Graduation punch by a fanatical terrorist group intent on destroying the school for some unknown reason, would have been right up their alley, and might have served to escalate and reenforce the activities of an already on site group instead of sending a hard core military team in to straighten things out.
(And I think I'm gonna stop there before this escalates from a post into a fan fiction. I obviously have just too much spare time on my hands lately))