He can go SSJ3. We've never seen it in the anime or any canon material, but he has demonstrated the ability to use it in video games such as Dragon Ball Fusions and Dragon ball heroes, etc. Search for videos about Super Saiyan 3 Vegteta and you'll find it. I just got done watching one that I couldn't paste here, but here it is as a thing you can copy and jump to.
Well I guess he can. Never seen it. Vegeta can reach SSGSS and the new SSGSS evolved. He could rach SSJ3 if he wanted to. He just doesn't, because his rage can push SSJ2 past SSJ3. But Toriyama has yet to say anything, so who knows?
Vegeta can definitely go Super Saiyan 3. He just never demonstrated it in the main story line of the Dragon Ball franchise because he never felt the need to use it since he has unlocked transformations that surpass that of Super Saiyan 3 though he has demonstrated that he can turn SS3 in various video games.
I do wonder something though. Are the Super Saiyan forms linear? I know that they have the name Super Saiyan 1, 2, 3, 4, etc. But they're named by our heroes as they reach them. Our heroes didn't even know what Super Saiyan was before. Remember when Vegeta mistook Zenkai and Saiyan spirit for Super Saiyan? Super Saiyan is the base for all of these forms which makes me wonder if all these forms aren't just grades. SSJ3 shares some similarities with SSJ 2nd and 3rd grade as it increases muscle mass and power while changing your look and consuming an intense amount of ki. SSJ4 is just Super Saiyan in a controled Oozaru form, which means it could easily be Super Saiyan 2. Yamoshi was said to only be able to use Super Saiyan in Oozaru form, which means if he was in control of Oozaru, then he would be SSJ4, which would technically be only his Second Super Saiyan tharnsformation. This is all speculation that may have gone a bit off topic, but I assume the answer to the question is this. Vegeta can use SSJ3, but doesn't because it's impractical. Just as he quit using Super Vegeta (SSJ 2nd Grade) when he discover Super Saiyan's mastered form and SSJ2. Maybe he couldn't before, but he deffinately can now, but he wo't because SSGSS is more practical. Just as he essentially skipped SSG until the Broly movie.
You need a high enough power level to become a Super Saiyan 4 according to Bulma and Vegeta in GT. I'm not sure Yamoshi would have that power level, he had no senzu beans or dragon balls, no gravity rooms or hyperbolic time chambers to help him in training so I think he probably barely got strong enough to become a SS and that was it since he already had his tail he went SS and after that went GGA but his power level wasn't high enough to become SS4 from there or we would have known about it in some media.
To be honest SS4 and GGA are the same thing. One isn’t stronger than the other rather one is more focused than the other. I don’t know about a power level required because somehow GGA just automatically uses the power of SS3 on top of GA, maybe it’s something to do with the maximum potential of the Saiyan? Not sure, but Yamoshi definitely could’ve done it, Saiyans back the could’ve been far stronger than the ones today.
When Bulma said that Vageta didn't have a high enough power level, she's refering to the time he had a tail. He only ecame a Super Saiyan AFTER he lost his tail, and Toriyama slready stated that you need suffencient S-cells AND a high enough power level. And as for SSJ4, It HAS to be more powerful than Golden Great Ape. Goku said that even Gotenks couldn't beat Baby. Using math, I determined that Fusion (Vegeto) is 400X stronger than the base form of one of the fusees. It's already said that Vegeto is as strong in base as SSJ3 Goku. Gogeta I assume would be at least close to that. Gotenks is weaker, but they have SSJ3 that they can use, so if Gotenks is 1/4th Gogeta/Vegeto then that puts Gotenks at a multiplier of 40,000X. If that is not powerful enough to beat Baby, but SSJ4 is. Then SSJ4 has to be more than SSJ3 X10 of Golden Great Ape.
That should AT LEASTput it on Super Saiyan Blue's level in my opinion. I know that it can't be true about Beerus' 70% PL from B.O.G, but I think he was also using a sort of grading system. In the anime, he says that SSJ2 Rage pushed him to 10% and was stronger than SSJ3 Goku. Goku SSG in the MOVIE is said to push Beerus to 70%. I know Beerus is stronger than this, but if he used the same grading system on both Goku AND Vegeta, then SSG must be 8X Stronger than SSJ3. Assuming that Goku made Beerus "Use about 9%" on King Kai's Planet.
SS4 is about 250x stronger than SS3 and 25x stronger than GGA maybe even more according to the information given comparing it to thw power a SS Fusion brings, as was shown in that thread. SS4 form itself is far, far more powerful than the fandom likes to say it is.
It's because people are biased by Dragon Ball GT and how "Horrible" it is. I personally liked GT and watch it regularly. The first saga "Black Star Dragon Balls" is not the best, but it mirrors original Dragon Ball perfectly. (Didn't quite care for OG Dragon Ball) but the Baby Saga and Super 17 and Shadow Dragons Sagas? Those had some of the best fighting I've seen in the entire Dragon Ball series! SSJ4 may be the ultimate form. It would only make sense if Gohan obtained it, since he's searching for the ultimate form.
Honestly I detest GT, it’s mostly for how annoying they made Pan and useless they made Trunks and the only slightly interesting villains were the shadow dragons, though this is my opinion. However SS4 is the only exception of GT that I really really like, it’s not that I believe GT is a terrible series and it’s form has to be as well, let’s face it, the way they handled SSR and SSB make them less than desirable forms. But Toriyama has a habit of making sure what he writes is the strongest version. And from previous comments you can see the math and reasoning behind my beliefs.
I suppose everyone is entitled to their oppinions on GT. It's not for everyone, but I'm glad you can see the awesomeness that is SSJ4. I don't even think Toriyama knows what the Multipliers are for the Super Saiyan forms. I think he just goes from the idea of "This form is newer, so it has more potential and is stronger than all others! And as for this form, since everyone likes it the most, we'll say that it stays relevant and get's stronger the more it's used!" Look at it like this. Broly in the original movies was beat by a squad of Super Saiyans and a Namekian. In the second movie, he was beat by a Super Saiyan 2. Now days, he has recieved no true change in his forms and yet he has the ability to go against two Super Saiyan Blues and even a SSB fusion form. Multipliers don't count anymore. I think it's looks and iconic-ness now.