Do I qualify to become an hourly shift coordinator at Burger King?

Ask the Burger King you want to work at. I would be surprised if a fast food place didn’t hire 16 years old. 

State requirements vary. Some states will allow a 14 year old to work.

From the BK web site

Generally, you must be 16 to work in an entry-level position, and 18 to work in any management-level position.

Burger king is like other fast food places.  You need experience

You need to contact the restaurant to see what they say. The reason you are getting different answers is due to different states having different age limitations. 

DON'T DO IT - I have friends that have worked at fast food places when they first left school, after a few months they were offered "promotions" to become floor managers or shift managers, they were paid a little bit extra and were given a set monthly wage but they were expected to work lots of extra overtime unpaid and unsociable hours and also turn up in the middle of the night to oversee deliveries, but the real management were just using them for cheap labour and had no plans to give them a meaningful promotion.

if you want to work in a fast food place then thats fine but just be a normal worker that way you can go home at the end of the night without all the stress