PWHole beat me to this but there are non-SSSI options below....
I don't think this is within a SSSI. The quickest way to check is the Defra Magic GIS website.
https://magic.defra.gov.uk/MagicMap.aspx
Screenshot of this attached- you open the Designations tab on the left hand side of the screen, then checkbox the land based designations, then statutory designations and SSSI units to refine your search. Lots of good info on this site. It also looks like its outside the National Park boundary. Thats going to restrict who you can ask for help.
You can also use Magic to see whether the land is within an agri-environment scheme. The taxpayer is paying for environmental outcomes in that case so there is a good chance that one of those schemes would prevent infilling of shake holes. The second screenshot here shows those schemes. There is a little info button on the top menu this will tell you the name of the agreement holder and how much public money he gets paid.
That feature is very useful if you want to find a landowner (though it will generally be a tenant or the farmer who has the agreement)
Back to the shake holes- if its in a scheme you could ask Natural England whether filing of shake holes was permitted within the scheme. If not you could ask them to do something about it.