The Book of Vile Darkness has received an update to its Oracle text. Specifically, the triggered abilities that Vecna gains are part of its copiable values. This means any enters-the-battlefield triggered abilities it gains (such as the one from Eye of Vecna) will trigger.

In the vast majority of cases, Vecna will gain the triggered ability printed on Hand of Vecna and the two triggered abilities printed on Eye of Vecna.

In some strange cases, you might make something into a copy of the Eye and/or Hand of Vecna and then exile that copy to activate The Book of Vile Darkness's ability. The ability that creates Vecna will look at the cards in exile, not the permanents on the battlefield, so it's possible to have Vecna gain different triggered abilities.