In this video, David Pawson asserts (at 29:20) that Ezekiel is a descendent of Zadok. I cannot find any verification for this. Is that true?
Is it true? There is as much evidence to confirm Ezekiel's Zadokite status as to deny it -- that is, none.
That Ezekiel is a priest is clear from the patronymic in Ezekiel 1:3 -
יְחֶזְקֵאל בֶּן־בּוּזִי הַכֹּהֵן
yĕḥezqēʾl ben-bûzî hakkōhēn
Whether that should be "Ezekiel ben Buzi the priest", or "Ezekiel the priest, son of Buzi" doesn't really matter - either way, Ezekiel is of priestly lineage, and so is himself "priestly". (Pawson's point that he is never "ordained" in order to enter priestly ministry is correct, however.)
What is not clear is whether he was "Zadokite" or not. As that Wikipedia article suggests, it is only the enthusiasm he shows for the Zadokites in Ezekiel 44:15-31 that makes this appealing. This is, in fact, the only "evidence" that Daniel Block adduces for the suggestion in his magisterial Ezekiel commentary (on p. 88 of vol. 1).
So, not something that can be asserted or denied: we simply don't know.