When Oscar Morales heard about the deadly flood in Kerr County, he had no doubt his team was going to assist with rescue and recovery.

Kristi Noem, the homeland security secretary, defended the federal government’s response on Sunday to catastrophic flooding in Texas that left more than 120 dead.

However, in Kerr County, where the devastation is most extensive, Kerrville Mayor Joe Herring said later he did not know what state emergency management resources were deployed ahead of time and that...