Normally, it is still possible to get married at church, even to someone who is not a catholic.
If the person is not a Catholic but a Christian, or a non-beliver, the celebrant will proceed in order to make the ceremony possible.
Sometimes reasons may interferre, which will make a church- marriage impossible. However, they vary according to circonstances.
For example, if a non Catholic person is against christian education provided for futur children, or if the person is against the meaning of a catholic marriage. But here is the moment for serious reconsideration.