Translation:There is cheese and oil at the supermarket.
Distribution doesn't matter. "There is cheese, and there is oil," is still a plural subject. Just because the sentence starts with "there" does not make the singular appropriate. Construct the sentence without the "there" and you'll see what I mean. You wouldn't say "Cheese and oil is located at the store," which is what the "there" amounts to. You'd say: Cheese is located at the store," or "Oil is located at the store," because those are both singular. However "Cheese and oil ARE located at the store," is the only correct way to say it once you conjoin the two subjects.
Check out what it is said in this link: https://www.grammarly.com/blog/there-is-there-are/
"Süpermarkette peynir ve yağ var." - There is cheese and oil at the supermarket.
Süpermarket "-te." This suffix means "at the."
Locative suffix "-te" describes location.
"Var" & "Yok" are adjectives that mean "there are/is" & "there is not." They are not verbs.
Why not "the supermarket has cheese and oil"??? - Correct other answer without "at the."
I'm saying your answer is acceptable but not according to Duo.
I note you said "spoken" English, which is not to say proper English. People speak all kinds of grammatical garbage such as "ain't", tense changes, and double negatives to name a few. Spoken English is what gives us "There" in the first place.
First, "there" is not the subject of the sentence. "There" is not a noun and cannot, therefore, be a subject. The subject is "cheese and oil," and it is a plural subject. The best grammatical construction would be: "Cheese and oil are at the supermarket." Granted, some people might say "Cheese and oil is at the supermarket," but that's simply bad English. It's like saying "They is at the supermarket," which is more clearly incorrect to most native speakers' ears (not that I haven't heard it - see the previous comment about the oft quality of spoken English). The confusion here comes from the fact that we are listing the individual ingredients which creates an impression that a singular verb (is) should be used rather than the plural verb (are). But the subject is plural and thus requires a plural form of the verb. Diagram the sentence if you need to, but it's a plural verb.
As you indicated, you could use the conjunction "and" to conjoin two complete sentences together instead of just the subject. That would be saying, "Cheese is at the supermarket, and oil is at the supermarket." In that case your subjects and verbs agree because you have two complete sentences. However, the translation Duo is providing is not in this format; it's in the plural subject format. And I think you'll agree that most English speakers would conjoin the subject rather than making the more awkward sentence that unnecessarily repeats "at the supermarket" twice.