Working through dispatch is possible at the age of 19. My friend, who was 19 at the time, started working for Servisair as a "Passenger Services Agent" - which I think is basically check-in. As soon as he got the job he enquired if there were any positions available in dispatch. After completing his PAX services training they moved him straight into dispatch.
All from a Summer placement and he still works the odd weekend there now. Although this was at EMA, so I don't know if it'd be possible at a bigger airport. Always worth asking the question though.