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).