Typically the officiant can sign your marriage license on the same day and make your commitment legally binding. However, there are situations when this is not the case, so it is best to check.
First the officiant needs to be licensed in the state/ region where the marriage is taking place. If he/she is not then you can still do a ceremony, it just requires that the legal element be managed separately (before or after the public ceremony). In such cases I think it is nice to do the ‘court marriage’ before your religious ceremony, so that everything is legitimate when you announce it to your family and friends.
It is also important to check each state and county rules. Different states have different rules regarding marriage laws. Some states require blood tests, others have timing restrictions, perhaps the ceremony must take place within 30 days of getting the license, or perhaps they have a required waiting period of 1 day.
Learn more about legal marraige and name changes here.