I hope you have already resolved your issue and I am including it here for other users.

you can use Chrome's Developer tool to check the Post command and this example below is what I'm currently using to control Recalbox through Homeassistant

These commands are for Recalbox 9.2 New Web ui

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rest_command:
emulationstation_reiniciar:
url: http://172.16.8.6/api/system/frontend/restart
method: POST
headers:
accept: "application/json, text/html"
user-agent: 'Mozilla/5.0 {{ useragent }}'
payload: '{}'
content_type: 'application/json; cha**et=utf-8'
verify_ssl: false

emulationstation_parar:
url: http://172.16.8.6/api/system/frontend/stop
method: POST
headers:
accept: "application/json, text/html"
user-agent: 'Mozilla/5.0 {{ useragent }}'
payload: '{}'
content_type: 'application/json; cha**et=utf-8'
verify_ssl: false

emulationstation_iniciar:
url: http://172.16.8.6/api/system/frontend/start
method: POST
headers:
accept: "application/json, text/html"
user-agent: 'Mozilla/5.0 {{ useragent }}'
payload: '{}'
content_type: 'application/json; cha**et=utf-8'
verify_ssl: false

recalbox_reiniciar:
url: http://172.16.8.6/api/system/reboot
method: POST
headers:
accept: "application/json, text/html"
user-agent: 'Mozilla/5.0 {{ useragent }}'
payload: '{}'
content_type: 'application/json; cha**et=utf-8'
verify_ssl: false