When I got married, I looked into only having a Church marriage and it wasn't possible. Getting married in the Church was automatically registered with the state.
This was in Canada. I have heard it works differently in different countries.
In Australia you don't apply for a marriage license. (I've only heard about them in movies).
When you go to the priest and he agrees to perform the marriage, you just go to the Church on the day with two witnesses, then after the ceremony the priest asks you to sign a Marriage Certificate, which is signed by the sopouses and two witnesses.
The priest then sends the relevant papers to the state as a record that the marriage was performed.
So we don't apply for (or demand) a marriage license.