Can same-sex couples marry online

If "through court" you mean civil marriage performed by a judge, then yes, but only in states and countries where same-sex marriage has been legalized. If you mean can they be married by court order, then this is another question. There have been cases where same-sex couples have sued for the right to marry and courts have ordered the state to permit same-sex marriage. Massachusetts and California are examples of this.

Another twist to the question would be the situation in most of Brazil, where a same-sex couple must first obtain a civil union, then ask for a judge to legally convert it to a marriage.