This sentence is complex, with prepositional phrases embedded in noun clauses embedded in an adverbial clause, but altering the punctuation can help clarify it:
And it shall come to pass (when the Lord seeth fit in his wisdom), that they shall minister unto all the scattered tribes of Israel, and unto all nations, kindreds, tongues and people. And [they] shall bring out of them unto Jesus many souls, that their desire may be fulfilled.…
A related difficulty is the apparently unattached noun clause: “that their desire may be fulfilled.” It is structurally parallel to: “When the Lord seeth fit in his wisdom that they shall minister… [and] that their desire may be fulfilled.… ” Although the phrase could relate either to the desire of the three or to those brought out of the nations, the next phrase certainly refers to the three: “and also because of the convincing power of God which is in them.” Therefore, it seems more reasonable that “their desire” also pertains to the three.
A related question is what, exactly, they desire; but presumably it is to bring souls to Yahweh, which would certainly be in character for apostles of the Lord.