You can carry
children around legally in the bed of a pickup truck all day long, the only catch is that if they're under 12 they have to be supervised. 20-135.2B(a) I've always found that amusing since if they're in the passenger compartment they have to be restrained.
There is nothing in state law that is dog-specific, but if the dog jumps out and causes a wreck, I suppose you could be charged with failing to secure your load, 20-116(g).