"Our listeners are what make our podcast possible" is grammatical. (But it took a little while for me to figure that out; thank you to everyone else who left comments and answers!) Like you, I felt uncomfortable with it after you brought it up, and I'll discuss the reasons for that below, but they are based on semantics rather than purely on the grammatical structure.
As a subject, "our listeners" triggers plural agreement on the corresponding verb in all varieties of English that I know of. In this sentence, that verb is "are." It isn't anything like a collective noun: collective nouns, such as "collection" or "group," are singular in form (morphology) but plural in meaning. But "listeners" is clearly plural in form, as it has the plural suffix -s. As you said, the answers to the question about 'Is it “5–6 weeks are a lot of time” or “5–6 weeks is a lot of time”?' only say it is possible to use singular verbs with nominally-plural subjects that are "quantities or measurements" (usually "of time, money, distance, weight"). There is no measurement involved in the noun phrase "our listeners," so I don't think it's natural to use a singular verb with it.
So that's what I'd say about the first verb in the sentence. But the verb "make/makes" is part of a later, distinct relative clause whose subject is the word "what." So to figure out which to use between "make" and "makes," we need to determine whether "what" is singular or plural here.
This is a tricky question, since relative pronouns always have one form, but can take different types of agreement depending on the situation.
A relevant situation that I have just thought of is the transparency of the relative pronoun "who" to person agreement, discussed in the following post: "You who is" OR "you who are". I think that I would not use person agreement in sentences like "Our listeners are what make [podcast name] possible." Here's what I mean. To simplify matters, assume we're talking to a single person, "Sally." I would say
Sally, you are what makes [podcast name] possible.
Sally, you are what make [podcast name] possible.
In fact, a Google Ngram search turns up no examples of the structure "you are what make"; "you are what makes" is attested, although rarely, and only consistently after the 1980s.
On the other hand, as FumbleFingers has discussed, it is true that "these are what make" is more frequent in the Google corpus than "these are what makes," and also, "they are what make" is more frequent than "they are what makes."
My interpretation of this is that in constructions like this with "what," the verb after "what" does not have to agree with the subject of the preceding clause.
However, it evidently can inflect for plurality. It does seem this is an exception to the default situation where "what" triggers singular agreement on a following verb. The English Forums explanation that FumbleFingers linked to in a comment
gives examples where a plural verb can/must be used if the complement includes a plural noun phrase (such as "What they desperately want are clothes and shelter"), but the relevant clause in the sentence you're asking about ("what make/makes [podcast name] possible") doesn't have a plural noun phrase complement. FumbleFingers' grammarphobia link also doesn't seem to cover sentences of this exact type. But perhaps it could simply be considered the inversion of "What make [podcast name] possible are our viewers," which is a sentence whose form is discussed (and confirmed to be grammatical) in the grammarphobia article. However, it doesn't seem safe to assume that we can invert the sentence and leave the verb the same.
This question seems to indicate that if a relative pronoun such as "what" starts a "free relative clause," the word "what" does not have to agree in plurality with the preceding noun "listeners." From what I can tell, it is a free relative clause in your sentence (I'm not even sure if bound relative clauses with "what" are possible in standard English; they usually use other relative pronouns such as "which" or "who"). This would mean that it's grammatically possible to treat "what" as either singular or plural; it appears that speakers usually do treat "what" as plural in this grammatical context (judging from the NGram that Fumblefingers has provided).
However, even if a plural verb is more commonly chosen, it does seem to be possible for "what" to be taken as singular here and get singular verb agreement. FumbleFinger's ngram link indicates that a notable minority of speakers take this option in the Google Books corpus, and I think the evident opacity of the structure in modern usage to personal agreement (as shown in sentences using the pronoun "you") provides some analogical support to the idea that it can be opaque to number agreement as well. When I first read your post, my reaction was that I would say "Our listeners are what makes our podcast possible."
Here are some other example sentences where I would use a singular verb rather than agreeing with the plural subject of the main clause:
- They are what is wrong with this place.
- They are all that makes my life worth living.
Different speakers may have different intuitions about this, however.
Maybe a good way to think of it is this: just as you could say "our viewers are the thing that makes our podcast great," you could also say "our viewers are the things that make our podcast great." Both are grammatically correct, although they may have slightly different shades of meaning. And since "what" can be plural or singular, it can mean either "the thing that" or "the things that."
An interesting point besides this: despite the fact that I would use a singular "makes" in the OP's sentence, I agree that only plural "make" is appropriate in a sentence like
Blue grass pastures, fields of clover; These are what make Mansfield
–"What Makes Mansfield Grow," from Around Mansfield by the Mansfield Historical Society
This is because I would not be comfortable saying "Blue grass pastures, fields of clover; These are the thing that makes Mansfield grow." Semantically, these are not one thing, or even one type of thing.