The speaker in this poem is issuing a direct address to God. You can see this in the opening line, in which he asks God ("O Lord") to use him, the speaker, as an instrument—specifically, a spinning wheel.

Throughout this poem, the speaker uses metaphor to imagine himself as God's useful instrument, a loom upon which God might knit his "twine" together and upon which the Holy Spirit might perform his necessary winding tasks. The speaker is appealing to God to make as much use of him as is possible—he is saying that, if God will only use him, God will be able to create more and better things through his willing servant.

In the final stanza, the speaker asks God to turn the fabric they have made together into clothing for the speaker's understanding, "will, affections, judgment, conscience, memory," so that when he finally appears before God on judgment day, he will be appropriately attired to meet his lord.

Edward Taylor writes in the first line, "Make me, O Lord, thy spinning wheel complete;" thus, the speaker is addressing God.  Taylor wrote many of his poems as meditations, meant to help him prepare sermons for his congregation, as he was a minister.  

In this poem, he begins by asking God to make him into a spinning wheel, asking God to work through him as a spinner would work with a wheel to make fine thread.  In the second stanza, he switches the metaphor to ask God to be the weaver, to make create a beautiful cloth that will:

Then cloath therewith mine Understanding, Will,

Affections, Judgment, Conscience, Memory,

My Words and Actions, that their shine may fill

My ways with glory and thee glorify. (ll. 13-16)

The speaker wants God to use him as a tool to both give God glory and will glorify the work that the speaker does (and, if one were to believe that Taylor is the speaker, that God with use Taylor as a tool to instruct and guide the people that he oversees in his congregation).