I can't give you a definitive answer to this, but I can speak from my experience.
People back then had very little proof of anything and were not expected to. Hence the number of marriages where the ages of the parties were distinctly off and the vast number of non-existent fathers named.
The proof of her widowhood would be the death certificate, which, if the vicar asked to see it at all, he would have returned. He may have made a note in some diary or day book but they are unlikely to have survived.
I very much doubt if he asked for any proof. If the parties were regular churchgoers and he had been around a while then he would have known the situation anyway.
There are documented cases of widows/widowers who were not who remaried in another town where their history was unknown and the chances of a Mr Rochester moment ("He has a wife still living") unlikely as divorce for the masses was not on.