In the Dragon Ball Super anime series, when Toppo used hakai he had to turn into God of Destruction mode. Now that Vegeta has been taught hakai, does this mean Vegeta now is able to turn into God of Destruction mode just like Toppo, or he just can use hakai?
They are separate things.
Toppo was able to use the God of Destruction Mode due to being the candidate to be the next God of Destruction.
But you don’t need to be a candidate to use Hakai, as demonstrated by Goku against Zamasu:
After achieving Perfected Super Saiyan Blue, Goku is able to perform the Hakai to a limited degree, however, it's much slower than Beerus', destroying Fused Zamasu progressively and requiring concentration throughout - thus Fused Zamasu takes advantage of the weakness by using Future Mai as a human shield, stopping the technique.
But yes, Vegeta can use the transformation, as demonstrated at the end of Dragon Ball Super #74: