The couple is responsible for filing for the marriage license. This is a point of confusion for many couples and first-time officiants. The officiant does not play any role in filing for the marriage license.
The officiant's role regarding the marriage license is to ask the couple to present it on the wedding day and to complete their portion of the license. Some states do require the officiant to return the marriage license. However, in most states the completed marriage license can be returned by mail.
Specific instructions on how to return the marriage license will be included when the couple is issued their marriage license. Make sure to read it carefully and follow the instructions.
For more information on what to do with the marriage license on the wedding day as the officiant, check out our page How To Fill Out a Marriage License.