The passage quoted is not one sentence. It is two sentences, conjoined by the word “with”. In the second sentence, the verb is tacit (i.e not expressed) by virtue of the conjunction. The second sentence on its own, with the verb explicit, would read “Instruction is carried out exclusively in the endangered language”.
That is not correct; it is one sentence. “with instruction exclusively in the endangered language” is a prepositional phrase attached to the main clause of the sentence. For it to be considered two sentences, one would need at least two independent clauses, and there is only one here. (And even if you had two independent clauses, it wouldn’t necessarily be two sentences, because you can have sentences with multiple independent clauses—this sentence is itself an example of such.)
OP Tipping is correct. Here is a more detailed breakdown:
Volunteer ‘apprentices’ pair up with one of the last living speakers of a Native American tongue to learn a traditional skill such as basket weaving, with instruction exclusively in the endangered language.
The subject of the sentence (i.e., who is performing the action) is the noun phrase volunteer ‘apprentices.’ The word apprentices is in quotation marks because they are not real apprentices, only acting as such for the purpose of this project.
The rest of the sentence is the predicate (i.e., the verb plus how and why the action is being done). The main verb phrase is pair up. English has many “phrasal verbs” like this, a verb and a preposition. In this case pair up means “join.” So the core or main clause of the sentence is Volunteer apprentices pair up. The rest of the sentence just adds explanatory detail.
With one of the last living speakers of a Native American tongue is a prepositional phrase that tell us who the apprentices will pair up with. (i.e., the direct object)
With instruction exclusively in the endangered language is a prepositional phrase that tell us how the activity will be done, i.e., what language will be used.
To learn a traditional skill such as basket weaving is an infinitive phrase that tells why they are pairing up, at least on one level. But the fact that apprentices is placed in quotation marks and the inclusion of the phrase about instruction in the endangered language implies that the true purpose of the activity is not to learn basket weaving, but for the apprentices to learn the endangered language. The basket weaving is only the means to the end. So there is some subtlety here. The stated purpose of the activity is not the real purpose.