It is fine to use "anybody" or "anyone" in your sentence.  It is grammatically correct to say:  There is not anybody there."  "There is not anyone there."  I agree with CocoPop that these sentences sound a little more relaxed and natural if we include the contraction "isn't" or "there's"...so usually people will say or write:  "There isn't anybody there" and "There isn't anyone there."   "There's not anybody there."  "There's not anyone there." 

The mistake with double-negatives will happen if you use "nobody" (or "no one") when there is also a negative word modifying the verb (is not, isn't, was not, wasn't, is never, was never).  These sentences are wrong: "There is not nobody there.  "There isn't nobody there."   "There was not nobody there."  "There wasn't nobody there."  "There is not no one there."  "There is never nobody there."  "There was never nobody there."

You can use "nobody" or "no one" if the verb is positive (for example: "is" or "was" without a negative word such as "no" or "not" or "never").  These sentences are correct:  "There is nobody there."  "There's nobody there." "There is no one there."  "There's no one there."

You can absolutely say "There is not anybody there.”

It would be slightly more natural to say either "There isn't anyone there." or "Nobody's at the gate."

What you couldn't say (because it is a double negative) is: There isn't nobody at the gate." Well, you could say that - but unless you're trying to (awkwardly) say someone IS at the gate, this sentence is the double negative.

Hi ... Just one Qs.

Yes, it is wrong.  It is grammatically incorrect.  It is better to use the following:

"Who said there was someone at the gate?  There isn't anyone there."

"Who said there was someone at the gate?  There is no one there."