Does “I don’t have either late-night nor between-meal snacks” sound natural?

Does the sentence sound natural?

I don’t have either late-night nor between-meal snacks.

No, it doesn't sound natural. You've mixed your correlative conjunctions.

Either is used with or and neither should be paired with nor, so you should choose from the following options:

I don't have either late night or between-meal snacks.

I have neither late night nor between-meal snacks.

Regarding specifically Does the sentence sound natural?, and assuming you mean in spoken English today.

It does not. You almost never say "nor" in every day spoken English. For that matter, you rarely say "either" in every day spoken English, and if you do you place it badly at the end.

A copywriter on a radio ad would probably write to be read:

"I don't eat late-night snacks, and I don't
eat between-meal snacks, either!!"

Furthermore! Hyphenated "slightly technical" phrases like "between-meals" do not sound natural in spoken English.

In real life (funnily enough, I don't eat snacks late at night and I don't eat snacks between meals) I'd probably say something like: I never eat snacks late at night and I never eat snacks between meals.