I have taught in 3 rabbinical and cantorial programs – and it has been my honor and privilege to participate in the ordination of Jews-by-choice. Most recently, in January 2011 I participated in the ordination of Shoshana Brown, a convert to Judaism, as a Hazzan and teacher in Israel.
In all the non-orthodox steams of Judaism there is no impediment at all for a Jew-by-choice to seek admission to rabbinical and cantorial programs. The famous letter from Maimonides to Ovadiah the Convert addresses the specific question of whether a convert might recite the words “Our God and God of our ancestors” in the liturgy, as well as the more general standing of converts. It is worthwhile to include a significant extract from this letter
'... Therefore, anyone who converts until the end of all generations, is counted amongst the disciples of Abraham our forefather and they are members of his household, … Thus, Abraham our forefather is the father of all worthy people that follow in his ways, and he is a father to his students, and they include anyone who converts.
Therefore, you should say 'Our God and God of our forefathers' for Abraham is your father. And you should say 'Who has given [the Land] to our forefathers' for the Land was given to Abraham.
As to the words 'that You have taken us out of the land of Egypt' or 'who has worked miracles for our forefathers' 'since you have entered under the protective wing of the Divine Presence and you share company with God therefore there is no difference between us and you, and all miracles worked, can be considered to have been performed for us and for you.' There is no difference whatsoever between us and you for any matter.
And certainly you should pronounce the blessing 'who has chosen us', and 'who has given us', and 'who has separated us', for the Holy One blessed be He has chosen you, and separated you from the nations, and given you the Torah, for the Torah was given to us and to the converts.
The spirit of this letter infuses the approach of many contemporary Jewish leaders and institutions who hold every door and opportunity open to Jews by choice.
It is worthwhile to note that there might be some 'waiting period' expected between the finalization of conversion and the application to a program that prepares a person for roles of Jewish leadership. As a convert becomes more engaged in their Jewish life and growth it may appear to them that becoming a rabbi or cantor is a natural next step - and this is not the case. Judaism has no 'religious orders' and every possible depth of Jewish life, growth, and learning is open to every Jew. The decision to seek to serve as a rabbi or hazzan is born out of a very specific understanding of personal goals and gifts, and is subject to careful exploration with the Admissions Dean of any Rabbinical or Cantorial School. It would not be unusual for a period of 5 or more years to elapse between the finalization of conversion and admission to training.
There are many Jews by choice who have graduated from the seminaries of the Conservative movement and who serve with great distinction as rabbis and cantors around the world, and this is the case with the other non-orthodox denominational streams, and also from the non-denominational programs such as The Academy for Jewish Religion, ALEPH (Jewish Renewal) and Hebrew College in Boston.
It is also important to note that there are some sources within the Halakhah that might be interpreted to preclude converts from serving as Rabbis or holding other leadership positions. These should best be addressed by my orthodox friends. It is also important to recall that the term 'rabbi' does not mean what it once did. According to Talmudic tradition the 'original' line of rabbinic 'ordination' was lost long ago and all contemporary 'rabbis' really hold the title only as a courtesy or modern convention.