But every time there's a thunderstorm around Marshfield, they say you can hear his rolling voice in the hollows of the sky.
Let's simplify the sentence to make it easier to analyse:
Every time there's a thunderstorm, you can hear his voice.
The phrase every time there's a thunderstorm is a temporal Adjunct (of frequency). Some grammars call Adjuncts "Adverbials". Notice that you can move this phrase around:
- They say you can hear his voice [every time there's a thunderstorm.]
In terms of what kind of phrase this is, it's a noun phrase. The head noun is the word time. This noun has a Determiner, the word every. Determiners are words like the, a, my, this and so forth. The word time is also being modified by a restrictive relative clause there's a thunderstorm. We could rephrase the sentence like this:
- Every time [that there's a thunderstorm], you can hear his voice.
- Every time [when there's a thunderstorm], you can hear his voice.
The word there is the Subject of the clause there's a thunderstorm. It is analysed as a pronoun in modern grammars such as The Cambridge Grammar of the English Language (Huddleston & Pullum, 2002). There are many reasons why. One reason is that it is repeated in question tags at the end of sentences:
- There are many thunderstorms, aren't there?
This is a different word from the locative adverb there that we see in sentences like I'll meet you there.
It seems then that the Original Poster is correct (apart from the small detail that every time ... is an Adverbial, but not an adverb - an adverb is always a single word).
Hope this is helpful!