Short answer: no. They are not GPS trackers and they don't send out their position.
Longer answser: Airtags work based on making a connection to a nearby iPhone which uploads the information so that Find My can see it. If there are no iPhones nearby then the airtag will not be seen by them and you will not be able to know its postion. So if your dog escapes and runs around the local farmers' fields then you won't be able to track it unless those fields are full of dog walkers, hikers and farmers with iPhones, but even then the position is approximate and not real-time.
Airtags are great if you misplace your keys or as peace of mind (or not) to know that your bike is where you locked it (they are not good for tracking stolen things because if the thief is carrying an iPhone it will tell them about the airtag so they can find it and bin it) but if you want real-time tracking of your dog then you need to buy a dedicated dog tracker with GPS.