Odds are that you will never need a weapon while on a chase, but after a storm hits, if you encounter looters (which, thankfully, doesn't seem to happen that often), it's probably better to have it than not to have it. Your best bet is to try to avoid situations where you might need to be armed. I'm former law enforcement and I very strongly support the individual right to carry. Having a firearm can actually prevent situations where you could be harmed altogether, without you ever needing to unholster your weapon. The one caveat that I would say for chasers/spotters is make sure you have a place to lock it up in your car when you're outside your car filming or observing, if you're not carrying it on your person. A lockable glove box is a great place. If you do get pulled over and you have a weapon in your car, telling the officer about it should be the first thing you do when the officer approaches your vehicle. Put your hands on the steering wheel where they can be clearly seen and inform the officer that you have a firearm inside the vehicle. Usually they will ask where it's located, sometimes they will pull you out of the vehicle for their own safety. If you're open and honest about this type of stuff, you will very rarely have issues, as the overwhelming majority of LEOs support the individual's right to carry.
I took first responder courses a number of years ago and I think this is a wise thing for all chasers to do. Hopefully you'll never end up in a situation where you need to render first aid, but if you are, it's better to have the training and be able to do it properly. Local fire departments, ambulances and police departments regularly take these courses, and if you ask, they would probably let you join in, especially if they know you're a storm chaser or storm spotter. The Red Cross also offers courses throughout the year in most cities, usually for free or for a very low cost. 99% of it is common sense, but it still is worth having the certification, even if you never need it. Chasers are sometimes the first ones on the scene, even before police and EMT/fire, knowing this stuff could save a life and that's ultimately why we do what we do.
