You don't really "need" anything to storm chase except maybe a thrill seeking personality.
If you want to do it well, having a great deal of capital would be a good start.
I will say though, driving through the Mid-west of the US and having a tornado form a hundred yards away and travel parallel to the highway you are driving on is quite panic inducing lol
It can be exhilarating, but it's also always extremely dangerous, even if you're an "expert."
Education isn't really required - I got into it by becoming a storm spotter for my county and relaying information to the emergency centers.
For my first 2 years of college, I went into meteorology and such. Back in 2008, I spent the spring & summer with a group of chasers through that outbreak. If I remember it right, there were something like 40-50 tornadoes in a 6-7 day period in my region alone, something like 400-500 tornadoes through the entire outbreak.
It can also be extremely boring. After that, we didn't see a single tornado for several weeks despite tracking several high profile cells.
Twister is pretty unrealistic, honestly. Fun movie, but you rarely get up close and personal like they did. At best you're 100+ yards away at all times.