Actually, not exactly. This has been very confusing to me when I learned Russian, because I expected Genitive every time when a sentence using Accusative got negated. And then I would get corrected to the Accusative form. To my understanding, Russian only uses Genitive for 'not having something" (У нас нет супа), but in other negations Accusative stays. (Maybe I still don't know some nuance). Polish on the other hand always uses Genitive when a verb needing Accusative gets negated.
I'm finding this little grammar book very handy (have it on kindle). Its gives the essentials, as it's name suggests.
It has the declension table after this quote, unfortunately they've cleverly made it uncopiable (its an image or something) so I can' just paste it in here. Anyway there are about seven pages of explanation after it lol.
"Regular Noun Endings Here is a summary chart of regular noun endings by gender. In many instances, there is a choice of ending, which is usually determined by the stem consonant (the consonant at the end of the word after the ending is removed). For rules on the distribution of endings, see below. A dash (—) means no ending."
Swan, Oscar (2008-10-12). Polish Verbs & Essentials of Grammar, Second Edition (Verbs and Essentials of Grammar Series) (Kindle Locations 415-418). McGraw-Hill Education. Kindle Edition.
Can we infer the speaker in this scenario? What I mean is: is this a more likely sentence to hear from a waiter, i.e. we do not have soup (available at the moment) or from the diners talking at the table, i.e. we are not having the soup (we are having the pierogi instead)?
Definitely more likely to hear from a waiter.
And you definitely won't hear Nie mamy zupy said to a waiter as in Polish mieć or nie mieć doesn't translate to anything connected to ordering meals (like the English collocation to be having sth) . It would sound more like we don't have any soup (there is also a chance the waiter will take pity on as not having any soup and bring us some). When ordering you can use wziąć (to take) instead. Eg. weźmiemy zupę (we'll take the soup) & nie weźmiemy ziemniaków (we will not take the potatoes).
Another common word said to a waiter would be poprosić (to ask for) used in future tense or conditional mood. Eg. poproszę pierogi (I'd like some pierogies) & poprosilibyśmy bigos (we would like some bigos). Sentences with poprosić instead of wziąć sound a bit more elegant and kind so you should these in a fancy restaurant.