Sorry, there is no storm chasing career field. The few individuals that do it are:
1. Chase tour operators. You may get hired as a driver. However this work is very seasonal and positions are hard to find. If your entrepreneurial skills are sharp and you have chase experience, you may be able to run a chase tour business yourself. However, again, the work is seasonal and the overhead costs stack up fast. It's only a career for those who put 100% into the business.
2. Scientific researcher. You have to have the right skills and education. Sometimes volunteers are accepted for chase work (such as in the IHOP project). However paid positions are limited to participating faculty members such as those of OU, as well as researchers at agencies such as NSSL. Your best chance of landing a paid job as a researcher is to work towards, at least, an MS degree in atmospheric sciences.
3. Photographers/journalists. You must already have a career in photography or journalism to get paid chasing, and of course you must be working for an employer who is interested in storms as an assignment (or you must be freelancing).
Tim