Yes, a Conservative or Reform Jew can be properly halachically married with an Orthodox rabbi overseeing the wedding. That said,the rabbi overseeing the wedding is required to determine whether both parties are halachically Jewish.
In the United States, where it is well known that Conservative rabbis do not generally sign off on a ketubah for an interfaith couple, it can be sufficient to present a parents' Conservative ketubah as proof of Jewishness, along with some proof that the mother is not a convert (since Conservative converts are rarely halachically Jewish).
In other countries, particularly in Israel, further documentation is often required for someone from a Reform or Conservative background to get married.
In these cases, the best option is to work with the Beth Din of America, which will provide halachic proof of one's Jewish (and single) status. The process can be done entirely online but takes around two months.
Source: I am an American from a Reform background whose parents specifically had a Conservative rabbi officiate at their wedding so there would be no doubt about their children's Jewish status, who got married through the Edah Charedit in Israel