It's pronounced Van Vote
In 1994, author Isaac Walwyn met Van Vogt at a science fiction book signing event. I've quoted him directly;
It was also at this time that I learned how to pronounce his name: he was telling someone that the "G" in "Vogt" was silent, making it sound identical to "vote." But by and large he did very little talking. He seemed very happy to be there, though, and when my turn came, shook my hand with a smile.
Additionally, Randall Garrett published a poem with the following title: "How to pronounce van Vogt" which was produced for the Omniumgathum, a vanity print of short prose and poems specifically intended to be read by members of the sci-fi writers community.
This poem was later reproduced in Isaac Asimov's Science Fiction Magazine, a magazine which regularly carried stories by Van Vogt.
Randall Garrett had met van Vogt on numerous occasions by this point and his poem clearly indicates the correct answer; Vogt to rhyme with wrote and mote (or as Wikipedia has it, /vænvoʊt/;).