This may very by state, as the state is the licensing body...I work in Texas, so I'll speak to that. I assume the process will be similar, but won't swear to it.
A misdemeanor on your record shouldn't prohibit you from becoming licensed, the problem would be an agency hiring you because of insurance concerns related to the reckless driving. If this gets expunged, I would think it's no longer an issue.
Question for you-Why do you want to be an EMT? If you have a degree in microbiology it seems like you could use that for a much better career with advancement opportunities. No judgment, I'm interested in your answer.