*Is it okay to omit the object?*
What object?
If you use *informing* in an object verb sense, then an object must be supplied or it is nonstandard grammatically and has no sensical meaning in any ordinary context.
It is almost certainly--or almost always--nonstandard or nonsensical to write or say *He left me without informing.*
If we deem *informing* to be a noun, it is difficult to think of a context in which the sentence would make sense. *He left me without purple* follows a common grammar pattern, but is also generally nonsensical.