You might want to check to see if there are any applicable county or city laws, but statewide, you can not shoot a dog solely for being on your property. Section 13A-11-241 makes it illegal to inflict any injury on a cat or dog, which would include shooting it. The exceptions to this rule are in Section 13A-11-246, which states that a person won't be charged with animal cruelty for shooting an animal if:
1.) They kill a dog or cat found outside of the property of the owner of the dog or cat when the dog or cat threatens immediate physical injury or is causing physical injury to any person, animal, bird, or silvicultural or agricultural industry.
2.) They shoot a dog or cat with a BB gun not capable of inflicting serious injury when the dog or cat is defecating or urinating on the person's property.
There are other exceptions, but those are the only two that would apply to shooting a dog or cat.