The midrash that I assume Rashi is quoting actually does say that she went to "מדרשו של שם ועבר". However, lower on that same page, it does identify Shem as the one who is the official navi (though not clear if it's talking about Rivka or Hagar). Interestingly, Abarbanel quotes the midrash as saying that she went to the beis midrash of Shem and Eiver, but it was Shem who told her the nevua.
Radak, however, writes that it could just as easily have been Avraham that Rivka went to. (See also the Abarbanel linked above, who writes that it is "more correct" to say that it was Avraham.)
As to why not Yitzchak, I would venture to say that we have no evidence that Yitzchak was capable of nevua at this point in his life (he lived one third of his life by this point). And even if he was, I'm not really surprised that she didn't go to him first. We don't find that Yitzchak and Rivka were the greatest of communicators. It seems evident that Rivka never told Yitchak what the navi told her about Eisav and Yaakov. Yitzchak and Rivka remained, according to the Torah, on completely separate wavelengths when it came to Yaakov and Eisav. But that is another question entirely.