If you simply amplify what comes out of a small AM transmitter then the resulting transmitter will be very inefficient. AM transmitters always modulate the final amplifier. There is no further amplification beyond the modulated PA stage.
Best option is to run the basic Tx at full carrier, then drive a final PA with either Audio via a transformer or PSU via a pulse width modulator and filter.
That said, for 10W carrier (40W peak envelope power) then something like a IRFP250 FET will happily do that much but will run quite hot., should give 10-15 dB of gain if matched properly