In order for most states to recognize the marriage as legally binding, the officiant must be physically present at the wedding ceremony. As the officiant, you must sign and complete the marriage license in person, along with any required witnesses, which must occur immediately after the ceremony.
This is because most wedding licenses require live signatures, as well as imposing requirements about where and when the ceremony can take place. If a couple has obtained their marriage license in Ohio, for example, and has their officiant registered to perform marriage there (as officiants are required to do in the state of Ohio), then the marriage must take place there, and the marriage license must be completed – in person – by all required parties. The officiant and the couple getting married must be physically present at the ceremony performed in that jurisdiction. Only then will the marriage be legally recognized by other jurisdictions.