Vegeta is one of the most famous characters of the Dragon Ball Universe. In fact, when Goku is remembered, there's always a chance that Vegeta will be mentioned too at one occasion or another. The Prince of the Saiyan race is definitely not happy to be the second most powerful Saiyan on Earth, as Goku is always one step ahead of him. But based on the way the "Dragon Ball Super" runs, Vegeta will also get what Goku achieved in another time (well maybe except Super Saiyan 3). And including in this list is the Super Saiyan God form.
According to Blasting News, Vegeta successfully achieved Super Saiyan God form in the manga, so there is a high possibility that he would also achieve the form in "Dragon Ball Super" anime. Well, it is not really that impossible since Vegeta is also a Saiyan with a good heart, despite the edgy personality.
In the anime and it manga, it is not still revealed how Vegeta got the form. Most of "Dragon Ball Super" fans argue that the form was achieved after Vegeta trained with Whis for six months. Whether Vegeta got it on his own or the rightful ritual was done on him, no one knows the answer yet, but the first one seems to be the likely answer. Vegeta is a stubborn character, so he will get anything that he wants in any other way.
But with Bulma pregnant, as reported by Otakukart, Vegeta will likely stay on Earth until his second child has been born. It is possible that Vegeta will be participating in the Tournament of Power, but he likely needs a lot of convincing. "Dragon Ball Super" shows Vegeta as a participant in the opening song after all, so no worries.
Super Saiyan God form is characterized by a Red aura and red hair appearance. The first to have this form properly is Black Goku with his "Super Saiyan Rose". Goku achieved the form through a ritual of the righteous Saiyans.