It seems like Vindolanda is kind of the Birleys' private gig, which is curious. Who is in charge of the archaeology there? Is it the state or some public authority or the Birleys themselves? (I read that they actually own part of the land!)
Perhaps as context it might be interesting to hear how archaeology is normally carried out in England, and if Vindolanda is any different?
This is a bit of a delay, but I have the answer to this one!
"Originally" Vindolanda was part of the Clayton estate. John Clayton, a Newcastle antiquarian, had bought up large parts of the Wall and related properties in order to preserve them - and also so he could excavate. On his death, it passed to his nephew Nathaniel. I have heard varying explanations for why the estate was eventually split up and sold at auction, but the story given in Whitworth and Leach's "Saving The Wall" is that Nathaniel's grandson, who inherited in 1928, lost an awful lot of money gambling and had to sell to be able to pay his debts.
Eric Birley was an archaeologist who had worked on Hadrian's Wall in the '20s under F.G. Simpson (father of Grace, the samianware specialist - the history of British archaeology in the 20th century is NOTHING if not a family affair). At the time of the auction he was excavating milecastle 9, but the prospect of having a site of his own was pretty exciting. He initially intended to bid for Carrawburgh, but on the advice of his senior excavator Thomas Hepple he decided on Vindolanda instead (which his father helped him to pay for). The next day the agents for the sale also offered him nearby Housesteads, but he was unable to make his money stretch that far. Birley is also significant for, as far as I can tell, responding to every major news report related to the Wall with some fairly dreadful poetry.
Vindolanda was officially handed over to the Vindolanda Trust in 1970, although the Birleys continue to be incredibly heavily involved as a family business - if you look at their staff page you'll see a lot of familiar names. This isn't incredibly surprising in terms of land use. There is a lot of archaeology in the UK and a good deal of it is on private land - another good case study is the London Mithraeum, which is essentially controlled by Bloomberg - and the rest is frequently controlled by charities, including the National Trust and English Heritage.
Archaeology as a whole has only fairly recently - within the 20th century - become a "professional" occupation. It is carried out in the UK by charities, private citizens, governmental bodies like Historic England, some dudes with metal detectors, etc. Although there are legal requirements - especially around the discovery of treasure, human remains or touching a scheduled monument - there is (quite reasonably) very little to stop you digging up your own land for the purposes of archaeology.
I'd recommend Hadrian's Wall: A Life by Richard Hingley and Saving The Wall by Whitworth and Leach for more details about the afterlife of the Wall.