Is this a grammatical sentence? Is it idiomatic for "Let's go?"
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It's colloquial, and might often be spelled, "Let's git," to reflect the pronunciation. It would be short for "Let's get going" or, as you suggest, "Let's go."
It isn't grammatical in and of itself because "to get" requires an object.
To "get" in English can be used to mean "to go", but usually in the sense that someone would shout "get!" to you-- "get" being short for "get lost".
So with that in mind, "let's get" could be interpreted as "let's get lost" which in turn could mean "let's go". But, it's definitely a slang sentence and not a complete one, and I've never seen the phrase used in that sense. Where did you encounter it?