Your question covers a number of different topics. Let me try to address them one by one.
First, I assume from your question (and I may be wrong about this) that you stand at the very beginning of your journey towards conversion. I welcome your desire to join us and, in the words of our tradition, to take refuge under the wings of the Shekhinah, the presence of God. While you speak of searching for a rabbi, I would encourage you to search for a community as well. Ours is a communal as well as individual identity, and most rabbis who work with Jews-by-choice stress to them the importance of living among the people, experiencing the ebb and flow of Jewish life together with them as an indispensable element of the conversion process. One can absorb a great deal of valuable information from reading books, learning Hebrew, and studying Torah. But all of that information becomes truly meaningful, certainly in its deepest sense, when one learns it and lives it as part of the people of Israel.
Second, although you are searching for an Orthodox rabbi to facilitate your conversion, I hope that you will keep yourself open to the entire range of Jewish religious options available to you. I say this not in order to turn you away from Orthodox Judaism, if you find Orthodoxy to be the most fulfilling Judaic path for yourself. I say it rather to emphasize that other options do exist and that those options might speak to you more persuasively and compellingly.
Finally, the literal answer to your question is “yes”: you most certainly can convert to Judaism in the United States under the supervision of a rabbi who is not “approved” by the Israeli rabbinate. As you note, the official Israeli rabbinate is an Orthodox institution, and its “approved” rabbis are Orthodox rabbis. Rabbis in the USA who represent the more liberal streams of Judaism – Reform, Conservative, Reconstructionist, for example – also offer programs leading to conversion. Indeed, the vast majority of Jews-by-choice enter our community under the auspices of these liberal rabbis. Unfortunately, the Orthodox Israeli rabbinate will not recognize these conversions as valid, because (like most Orthodox rabbis generally) it does not accept the liberal streams as authentic versions of Judaism. We liberals disagree, of course, but there the matter stands. On the other hand, the government of the state of Israel, as a secular entity, will recognize your conversion as valid, and, as a Jew, you would be eligible for citizenship under the terms of the Law of Return. This wouldn’t solve all of your Jewish status problems, though, since personal status laws (i.e., marriage and divorce) for Israeli Jews tend to be under the supervision of the rabbinate, which unlike the civil government would not acknowledge your conversion. Still, these problems may not be insurmountable. When the time comes, you should consult a liberal rabbi in Israel for guidance.