As far as English Bibles are concerned, I have only the NRSV and NIV at hand: both of them include the "me". Most recent French translations do.
I will quote Metzger (A Textual Commentary on the Greek New Testament): "Either the unusual collocation, "ask me in my name" (yet it is not without parallel, cf. Ps 25.11; 31.3; 79.9, where the Psalmist prays to God for his name's sake), or a desire to avoid contradiction with 16.23, seems to have prompted (a) the omission of me in a variety of witnesses (...) or (b) its replacement with ton patera [the Father]... The word me is adequately supported (...) and seems to be appropriate in view of its correlation with egô [I] later in the verse."
I skipped the manuscript references but the me is in the older ones (P66, Sinaiticus, Vaticanus etc.). As usual in textual criticism, the question is: what best accounts for the difference? The position of most modern scholars is that it is best explained as an omission than as an addition.
(Btw, the me is absent from the so-called "majority" = Byzantine tradition, which explains why older English Bibles do not have it.)