Can I sue for someone failing to act, or an omission, as negligent in a personal injury lawsuit?

In Florida, yes you can. For example, if an employee of a retail store fails to follow protocol or the rules of that store, and doesn't inspect the produce aisle and something is left on the floor and a person is injured, or if the employee doesn't follow the rules that, if there is a glass that is broken on the floor or there is water or syrup that's on the floor, and let's say that retail store says the employee must stand there until another employee can obtain a mop to clean and make this area safe, in those types of situations, the injured person has a claim and can file a lawsuit against the retail store for the negligence of its employees.