In the anime, Vegeta went straight to Blue after training with Whis without any explanation.
In the manga, Vegeta goes to Super Saiyan God in a battle with Black Goku to Goku’s surprise. It appears that the order of God before Blue will be more reasonable because in the battle, Vegeta makes good use of two states: He uses God to save long-term fighting power, but when the time comes to counter-attack go to him and Blue to enhance speed and power.
So Vegeta can have God in the following two cases:
Saiyan Beyond Gods
This article is more about this state. Super saiyan Blue is considered God’s Super Saiyan, and to use these states Goku and vegeta always need God Ki => This state is called Saiyan Beyond Gods
That is, their God Ki will replace Ki often. So for the range of opponents, they use Ki often on Ssj 1-2-3 (vegeta is not), and strong opponents go to God and Blue.