Here is a sentence:
“Are there other animals vanishing that, initially, may not seem essential, but without their activity, we couldn't envision life?”

The word "but" connects two parallel ideas or clauses. The two words in parallel are:

(1) "may not seem essential"
(2) "without their activity, we couldn't envision life"

So my question is;
If this is true as mentioned above, does “that” in (1) function as a relative pronoun, while “that” in (2) functions as a conjunction that introduces a subordinate?

Question

Here is a sentence: 
“Are there other animals vanishing that, initially, may not seem essential, but without their activity, we couldn't envision life?”

The word "but" connects two parallel ideas or clauses. The two words in parallel are:

(1) "may not seem essential"
(2) "without their activity, we couldn't envision life"

So my question is; 
If this is true as mentioned above, does “that” in (1) function as a relative pronoun, while “that” in (2) functions as a conjunction that introduces a subordinate?

Answer

"But" is a coordinating conjunction, and you've already identified the clauses that it's connecting.

"That" is functioning as a relative pronoun to avoid repeating "animals."  To see it clearly simplify the structure, change the question to a statement and don't include the relative pronoun:

There are other animals. The other animals are essential.