Hi Masato,
No, it can't work without Amazon Alexa server.
The works that RT106A take is
Far-field audio front end softDSP
Acoustic echo cancellation
Ambient noise reduction
Beamforming
Barge-in
Playback processing
Codecs
Wake word inference engine
Media player/streamer
MQTT, lwIP, TLS
Discovery and onboarding
All drivers, including Wi-Fi® and Bluetooth
By the block diagram in MCU-based Solution for Alexa Voice Service | NXP , you can clear get which part will be realize by RT106A.
Regards,
Jing