Yes, definitely expedite the birth certificates by mail. Even so, they are cutting it very
close as it may very likely take more than a few days for the county to locate the birth certificate, process it, and mail it. See average processing times HERE
Additionally, they will also need photo identification from a government agency -- driver license usually suffices. If they are not drivers, they may be able to get a non-driver photo ID from the motor vehicles department. Neighbors of mine just did that.
Also critical -- In addition to needing these documents at the border crossing, they will also need them to board the ship.