There isn't a turnkey solution I know of, but if you can have a tool save voicemails into a folder, you could probably have a script generate text files based on a Speech To Text (STT) solution:
Ahh, so this isn't a processing issue it's a data access issue.
Frankly, if you can't access the raw data of your voicemail inbox, probably no third party developer can too. This means that the only way to implement such a tool would to be to work with the voicemail provider. If they're a for-profit company, they probably have no incentive to make the data available unless there's a big moneybag involved somewhere in the exchange. That's probably why no such tool exists.
Many phone/cell providers provide a method to send voicemails to a third party, if you setup call forwarding (busy or unanswered, don't set unconditional) for reference, this page covers how to do that for T-Mobile