If, for instance Sirius Black happened to turn into a ghost when he died, would he have been able to turn into his dog form as a ghost? Also, if Remus were to turn into a ghost, would he turn into a werewolf-ghost at every full moon?
While it appears that ghosts aren't capable of doing magic, it isn't clear whether this applies to magic that affects only them or is simply a consequence of the more general rule preventing them from fully interacting with the world.
Even if they are unable to perform magic, even upon themselves, that does not actually rule the Animagus transformation out, since according to the wiki it is not actually a spell:
The difference between Transfiguration and the Animagus transformation ability is that an Animagus can change into an animal whenever they want, without a wand or an incantation. Being an Animagus is an ability, and Transfiguring requires a spell.
Similarly, lycanthropy is a magical illness. Given that injuries (such as having your head nearly cut off!) do persist when you become a ghost, it seems plausible that illnesses, even magical illnesses, would persist as well.
Ghosts are unable to learn new skills but retain the skills they already possess. So while a ghost certainly could not become an Animagus if they were not one in life (and would presumably also be immune to lycanthropy) there is no obvious compelling reason why they could not retain the ability after dying.
In the absence of a direct canon answer, I think "maybe" is the best we can do.
My personal guess would be that it would depend entirely on the ghost's own expectations; if they expected to be able to continue to take their Animagus form, then they would be able to do so.
In Harry Potter (and most other stories involving ghosts) the ghosts are only a pale imprint of the person at the moment they died.
They never change at all in any of the books: the bloody baron still wears the bloody clothes he wore when he killed the white lady. And she still has the blood stains on her chest (did not look for the exact quote). Nearly Headless Nick is -well- still nearly headless and also still wesrs the old clothes from his execution.
My imression is: they stay exactly in the state they died.
As a conclusion I‘d say: if an animagus died in his animal form and came back as a ghost, he would still be an animal and not be able to transform back (and vice versa), and same would be true for a werwolf...