A minor thing, but Balgruuf is, according to Rikke, refusing the Legion's right to station troops in his city. A circumstance surely brought about by the fact that forcing the issue will likely drive him to join the rebels.
That said, each Jarl(and Count in Oblivion) have their own forces loyal to them/their Hold/County. They are not part of the Legion, for the most part, with some exceptions. Like Captain Aldis, who is apparently both captain of the Solitude Guard and an officer in the Imperial Legion, which has its headquarters in Skyrim in Castle Dour. But the Solitude Guard is not part of the Legion inherently, because Tullius has to conscript them, according to a conversation between Elisif and her Housecarl.
Other things that puts the local guards as not part of the Legion normally is...
... Ocato saying the counties in Cyrodiil are on their own because he doesn't have Legionnaires to send them.
... Balgruuf's steward asking him "What is the harm of letting some Legionnaires dying instead of your own men?"
... the Holds that side with Ulfric still having guards. The rebellion started on the large scale, at the earliest, some months prior to the game starts. Not a lot of time to beat down the Legion in those Holds and then build a guard force back up.
There was a time during the Akaviri Potentates in the 2nd Empire when the vassals were not allowed to have their own armies, but this appears to not have been a practice under the 3rd Empire.
As for autonomy, we know that the nobles in Skyrim, at least, won some autonomy from the Empire after the War of the Red Diamond, as payment for siding against Potema after her only son died and she became a succession crisis in making. Now, that was 500 years ago, but seems to have been maintained as a status quo ever since, so long as Skyrim didn't rebel.
The Legion undoubtedly has a presence in every province(less so as of 4E201, because of the Thalmor threat to the south) most of the time, but it seems they remain largely hands-off when it comes to the upholding of local custom and law. They control certain places directly(like Ebonheart at the time of Morrowind), while others they maintain a presence to sometimes lend assistance to local forces(most larger settlements in the game Morrowind).