They are on the list of protected birds of the Federal Migratory Bird Treaty Act and it is illegal to hunt them unless you have a special permit to do so.
U.S. Fish and Wildlife Service., Interior § 10.13 List of Migratory Birds.
Woodpecker:
Acorn, Melanerpes formicivorus
Black-backed, Picoides arcticus
Downy, Picoides pubescens
Gila, Melanerpes uropygialis
Golden-fronted, Melanerpes aurifrons
Hairy, Picoides villosus
Ivory-billed, Campephilus principalis
Ladder-backed, Picoides scalaris
Lewis’, Melanerpes lewis
Nuttall’s, Picoides nuttallii
Pileated, Dryocopus pileatus
Puerto Rican, Melanerpes portoricensis
Red-bellied, Melanerpes carolinus
Red-cockaded, Picoides borealis
Red-headed, Melanerpes erythrocephalus
Strickland’s, Picoides stricklandi
Three-toed, Picoides tridactylus
White-headed, Picoides albolarvatus