There is no indication in the text of any early English translations that the use of thou/thee/thy/thine signifies "informal you". That notion is a theological and interpretative one, and frankly better asked about elsewhere (SE: Christianity or SE: Biblical Hermeneutics).
For instance, Jesus addresses Satan with Thee: are we to take this to mean we (or even he) was using "informal you" rather then simple singular you to address Satan?
Language-wise, the use of thou/thee/thy/thine exists in the English-language translations of Wycliffe (1384), Tyndale (1526), The Great Bible (1539), the Geneva Bible (1560), the Rheims New Testament (1582) and somewhat belatedly, the King James Version, or KJV, (1611).
Thou/thee/thy/thine were used in these translations to preserve the distinction in number between you-singular and you-plural, which is also found in Latin and Greek (I am not sure about ancient Hebrew), the languages that these translations are based on. Wycliffe had only the Latin at his disposal; the Rheims (the official Roman Catholic translation) was also based on Latin; while others, including the KJV, were based both on the Greek and on earlier English versions (thus, the translators of the KJV consulted not only the Greek but the Rheims, etc).
This distinction in number can be seen in such passages as Luke 21:31-32. For example, the KJV has:
31 And the Lord said, Simon, Simon, beholde, Satan hath desired to haue you, that he may sift you as wheat:
32 But I haue prayed for thee, that thy faith faile not; and when thou art conuerted, strengthen thy brethren.
KJV (original spelling, 1611) (link)
You does not refer to Simon alone, but to all the disciples; while thou/thy/thee refers to Simon alone (you-singular).
As another example, note that the use of singular-you is also used in the exchange between Jesus and the Devil in Matthew 4 (the three temptations). See for example Matthew 4:9-10:
9 And [the Deuill] saith vnto him, All these things will I giue thee, if thou wilt fall downe and worship me.
10 Then saith Iesus vnto him, Get thee hence, Satan: for it is written, Thou shalt worship the Lord thy God, and him onely shalt thou serue.
Screen shot of original 1611 KJV manuscript of Matthew 4:7-10:
Here is a screen shot of the Our Father from the Douay-Rheims Bible of 1610, which includes the 1582 Rheims NT:
And a screen shot of the Our Father from the King James Version of 1611.
If you want a theologically-based answer (for instance: how should people "address the creator of the universe"?) from a Christian point of view, you should probably post this on SE: Christianity or SE: Biblical Hermeneutics.
Note: I favour the OP's title Why does the Our Father use “thy”? rather than the edited version If “thy” is an informal pronoun, then why does The Lord's Prayer use it to refer to God?
Also, my suggested edit to the OQ, adding the tag grammatical-number remains apropos and should not have been rejected.