I was wondering if a convert (someone who was not born a Jew) may, by the rules of the state of Israel, migrate to Israel (make aliyah, as defined by the State of Israel)?
Yes, the only limitation is that the conversion is recognized by the Jewish Agency and the Ministry of Absorption. Having your conversion recognized by the Ministry of Absorption can be either easier or harder than having it recognized by the Chief Rabbinate of Israel depending on the situation.
Israel's Law of Return says that
and defines "Jew" as