Thanks @Muzzamil Anwaar for sharing this. Does your VPC ACL also allow access? I think by default it should but just wanted to ensure the request is going out of the instance first. Also, if you can try to use the speech to text API using curl to check if all the ports are open from the instance does the request go through? A simple request to get the token should let us know if this works and debug further.
curl -v -X POST \
-H "Content-type: application/x-www-form-urlencoded" \
-H "Content-Length: 0" \
-H "Ocp-Apim-Subscription-Key: YOUR_SUBSCRIPTION_KEY"