Yes! Permission should be requested in advance of performing any cemetery preservation work. Depending on the location of the cemetery this may include many different situations. Cemeteries that have staff on the sight should be notified in advance of work being performed. If a cemetery is maintained by the town then there will be a contact number available for you to reach.
Abandoned cemeteries represent a unique situation where there is no one simple answer. It would be wise to contact the local historic society, or funeral directors in the region may have information to help determine if there is any stewardship of the site.
Do I need permission to work on my family's plot?
This all depends on the location of the specific cemetery, and the regulations that are applicable. Generally speaking, cleaning and working on your own family or relative's gravestones should always be allowed.