Sephradim say "she'Atah". And there are other cases of "sha'Atah", eg in Emes veYatziv.
In the Torah, you will not find a "she-" prefix. HQBH uses "asher". (Nor the "kishe-" for when / whenever.)
In early Navi, you'll find "sha-". Not too often, but one case is in Shofetim 6:17, when Gid'on refers to Hashem as "sha'Atah". (Another is the two occurrences of "shaqqamti" in Shiras Devorah, 4:7.)
Joshu Blau of the Academy of the Hebrew Language says that this was the Northern contraction of "asher", but the Southerner's "she-" eventually wins out. (Phonology and Morphology of Biblical Hebrew, pg. 183) Except that Devorah was in Bet-El, so unless she borrowed northern coinage to make the poem work...
Tefillah used to tend toward Mishnaic Hebrew in both Ashk and Seph. With exceptions like the masculine "lakh" in "Modim anachnu Lakh". But when the printing press made publishing a siddur with nequdos possible, some hypercorrections went into Nusach Ashkenaz by experts convinced we're all saying it wrong.
These tended to be maskilim, as few else in Ashkenaz were studying diqduq. One prominent name is R' Shelomo-Zalman Hanau (Razah). Research seems to indicate his diqduq rules were employed by Lubavitch's Alter Rebbe in making Nusach Ari. But that has been debated here in the past. In any case, somehow, people managed to buy into the idea of changing large chunks of the vowelization of their davening in a comparatively short time.
Although, the medieval manuscripts indicate that we were using Mishnaic Hebrew all along.
These corrections made the Ashk siddur a lot more biblical. It began the debates between "morid hagasham" vs "morid hageshem", since in Mishnaic Hebrew there is no "hagashem", even if it's the last word of the sentence. And in earlier Ashkenaz, they said "vesein chelqeinu besorasakh, sab'einu mituvakh" -- just as Seph still say.
The presence of "sha'Atah" in Shoferim meant that that became the form in Ashkenazi in the past 2-3 centuries. And indeed, in earlier manuscripts, the word is "she'Atah".
In addition, it is possible that the "sha-" is the usual contraction for when one word is taking both the "she-" and "ha-" prefixes. That Gid'on was calling G-d "The You", and this is what we're imitating in davening.
