Is there a word for someone who doesn't necessarily steal anything but would compulsively snoop through someone's things if left alone, similar to kleptomaniac (someone who would compulsively steal things)?
The word snoop also has a noun form:
: to look or pry especially in a sneaking or meddlesome manner
: one that snoops
So, somebody who snoops is a . . . snoop.
I haven't been able to find a term that's as technical as kleptomaniac is for a compulsive thief. A technical phrase might be "pathological nosiness" or "pathological snooping".
As Jason Bassford notes, you can use snoop as a noun here. There are also idioms and slang in particular dialects, such as nosy parker.
It's also a feature of relationship obsessive compulsive disorder, but it exists independently of that diagnosis, and is only one feature of it.