Currently, while there's not an out-of-the-box IVR component to handle this need, options do exist. It's also evident there is interest in the topic. Here's what I was able to compile for you.
The simplest solution would be leveraging the mail handlers to process vm attachment files to tickets: https://confluence.atlassian.com/display/JIRA060/JIRA+as+a+Support+System#JIRAasaSupportSystem-Phone. Each project using JIRA Service Desk can have it's own mail handler to create tickets, so you can route email attachments to different projects depending on where the issues should be created. More about JSD and email: https://confluence.atlassian.com/display/SERVICEDESK/Setting+up+the+email+channel.
Or a more integrated setup would require customization via the API but results seem promising. You can check out more at this question:
There may be a more streamlined process if you're already using Twilio:
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