I don't know about Amazon, but Google does have Cloud Speech API; seems to be what you're looking for:

https://cloud.google.com/speech/

If you have a podcast, at https://www.podigee.com we're about to launch a beta transcription service that is integrated into the podcast publishing workflow. Drop me a line if you want a beta invite for the feature.

Amazon Lex: https://aws.amazon.com/lex/