If a couple is married by the civil authorities in Mexico, do they need a marriage certificate to get married by the church in the US?

The church will let you know that. However, once a couple is married legally in a different country, they are considered to be married i the US too.

The couple should consult with the specific church in the U.S. to find out its requirements for a church wedding, but for immigration legal requirements, not that generally if a marriage is recognized as lawful in the place where it takes place, the USCIS will regard it as valid for U.S. immigration application purposes.