I'm a an EMS newbie, but prior to joining the ranks here I'm a volunteer for my counties EMA, as an EOC worker, emergency communications and several other roles. For the most part, disaster's have binders on the shelf, each one has a plan for who will do what and when. As someone else has already mentioned they almost never get touched except for drills and updates.
As far as EMT's you will do the same thing you always do, there will be things like opening shelters, and whatnot but that won't be at the emt level, thats up to public health and red cross and other departments. The EMT role in a tornado disaster is the same as if you had a big MVC you will respond like normal, there will be triage, take the high priority to the hospital rinse repeat. About the only thing that might happen differently is if you weren't working you get called in to assist, but only after something happens. They hardly ever call anyone in ahead of time, unless your one of the county storm spotters or something.
It costs too much to mobilize anytime there is a chance for tornado's, especially in illinois down to oklahoma or anywhere else in tornado valley. That's why us volunteers jump in because were free and realize the importance of being prepared. So unless your ems allows you to show up to work for free just-in-case and it doesn't violate union laws or anything else, as I said before nothing is likely to change until something happens, and 95% of the time you will be doing the same thing you do now.