I know that stormtroopers did a significant amount of policing for the Empire, but surely they could not have been the only organisation providing law enforcement.
Who acted as law enforcement in the Galactic Empire (apart from stormtroopers)?
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Local (planetary) police forces served as law enforcement on many systems in the Empire. An example in canon is the Coruscant Security Force, which was composed of police droids:
as well as non-droid officers:
Another example from canon is the Royal Naboo Security Forces, an organization which we know existed during the Imperial era because it appeared in Shattered Empire, Part II which took place after Episode VI Return of the Jedi. Many of these police forces had existed since the Republic. The Empire also had a secret police force, the Imperial Security Bureau. The Imperial military (including stormtroopers) acted as law enforcement on more rebellious planets such as Ryloth (where the Empire used slave labor).
One example from Legends at least is CorSec, or the Corellian Security Force, which acted as the primary police force for Corellia with an imperial intelligence liasion officer to keep an eye on them.
It is important to note that security forces were planet-specific. For example, we have: Coruscant Security Force, Royal Naboo Security Forces, Telos Security Force (TSF).
All the planet specific security forces were more or less obedient to the Empire (Imperial Security Bureau and/or Imperial Intelligence).
Additionally to these, on many planets there were other private security forces (like the Security of the Exchange or Czerka Corporation) also mostly compliant to Imperial rules and were pretty powerful on the locations of interest. So in case the local forces were resilient to comply with what's needed, the private ones made sure that thing go a little more the Empire's way.