Yes. But in formal English, it means something slightly different.
"I wish I went" means I do not go to Harvard, and I don't have the opportunity to go to Harvard, but life would be much better if I could go to Harvard.
"I wish I would go" means that I do not go even though I have the opportunity: I am lazy, and this is frustrating.
Obviously, you wouldn't normally say that. You might say something like, "Steve has so much talent, and he wastes it all -- I wish he would go to Harvard!" Or: "Jane has been here for four hours now, and she's so boring -- I wish she would just go home!"
However, I have noticed some native speakers saying or writing "I wish I would go" when they mean "I wish I went". I don't recommend you do the same, but some people do use this construction. The context will usually make it clear what is actually meant.
Student or Learner