There are things that we say in davening in Aramaic. I have seen a few explanations as to why (as per this Q and A) but I don't know where Ushpizin fits in with that. I'm not sure if anyone would call Ushpizin a "prayer" so I don't know if its language is up for discussion as other things' are.
Unlike Ha Lachma Anya (are parts of the Haggadah considered "prayers"?), we don't need the common man to understand this invitation as it is aimed at the souls of biblical figures -- are we assuming that they don't know Hebrew? Are we saying something that we don't want angels to understand? Or is this just a case of "it was written in a text which was in Aramaic so, whatever"?