I know that it's probably best to assume that he did, since he reached SSJ4 in GT and SSJB in Super, so he probably achieved it at some point, but is it ever stated or implied in guidebooks, Super etc?
I mean you can construe Vegeta's statement at the end of Z to imply that he did (he said something along the lines of "beating Kakarot" if I recall correctly). Other than that, there's nothing that implies it and no supplementary material that says he did, unfortunately.
As far as know, he has never achieved Super Saiyan 3. Most likely because he recognized the drawbacks of the transformation and decided to train with the more controllable Super Saiyan 2 transformation.