We have no record of Elijah anointing anyone. This is not to suggest that he did not anoint anyone but it is not recorded in Scripture. On this verse, the Pulpit commentary has:
In the Hebrew the time of the anointing is indefinite. This commission
has long been a crux interpretum. For neither Hazael, nor Jehu, nor
Elisha, so far as we have any record, was ever anointed by Elijah.
Elisha was called by him to the prophetic office. Hazael, it is barely
possible, may have been anointed secretly, like David (1 Samuel 16:2,
13), but all that we gather from Scripture is, that he was called in
an indirect way, and certainly not anointed, by Elisha (2 Kings
8:12-15). Jehu was certainly anointed, but it was neither by Elisha
nor Elijah (2 Kings 9:1, 6), but by one of the sons of the prophets.
All we can say, consequently, is that the command was obeyed in the
spirit, and no doubt in the best possible time and way.
Thus, the answer to the question is that we have no record of Elijah anointing: In fact, his successor anointed some but not Elijah. Further, the record in 1 Kings 19:19 is NOT strictly of an anointing (with oil) but merely an appointment. The same pulpit commentary above offers some further remarks:
… we can readily understand that the word "anoint" was, as in Judges
9:8, Isaiah 61:1, never meant to be construed literally. For in the
first place, we have no record elsewhere of the anointing of any
prophet; and secondly, it is remarkable that when Elijah might so
easily have anointed Elisha, he did nothing of the kind. It is clear,
therefore, that he understood the word to mean "appoint." And the root
idea of anointing, it must be remembered, was the setting apart for
the service of God (Exodus 29:6). Hence it was (Bahr) that vessels
(Exodus 30:26 sqq.), and even stones (Genesis 28:18), were anointed.
And when we find that these three persons were set apart sooner or
later, and in different ways, to fulfil the high purposes of God, that
ought to suffice us.
Sometimes we must be satisfied with the Bible's way of using words and its incomplete history of events.