This isn't really about a tag, per se, it's more a question of how to provide audio in a question in a similar way to how images are already provided for. Note that images are actually hosted on a StackExchange-specific sub-site on Imgur, and that is fairly seamlessly integrated into the editor to allow a direct upload which is stored on Imgur and its link added to the posting.
You're basically asking for a similar integration for audio. There IS one, of a sort, for SoundCloud (or for YouTube): a per-site configurable Embedded audio player which, with community support here, presumably could be enabled for our site as well.
Note, as the handling of images should show, that at the end of the day you're uploading content to some hosting service and what ends up in your posting is a link to that content. So if a suitable content hoster exists, you can do the uploading yourself and then include the link in your posting manually. If the embedded player integration is enabled and you've included the link, on its own, as its own line in your post, it would be replaced inline with a player widget. This provides at least one possible avenue for including audio or video more seamlessly into a puzzle. Even without the embedded player, or for content hosted elsewhere, you could still do the same thing, but playing the content back would require the user to click an external link and be taken away from the puzzle posting, at least temporarily.
Where should binary files be stored? already discusses how/what off-site resources should and should not be used. Audio data isn't covered there, but hopefully the general idea comes through.
I'm not familiar with many audio hosting services, but SoundCloud is already explicitly supported on StackExchange sites where the embedded player is enabled. In addition, this (quite old) Webmasters.SE posting The YouTube of Sound gives other suggestions in the answers, some of which I'm sure are still relevant. See also http://www.audiohostings.com/ for other alternatives. There have also been other puzzles which shared content on, e.g., vocaroo and YouTube, as well as some more content-specific sites for, say, midi files.
Specifically what you'd be looking for is a reputable site that ...
- isn't likely to disappear any time soon
- lets you upload a file that can be played back through the website in question, rather than having to be downloaded locally and run through some other program to play back
- is freely usable by viewers without needing to register/sign-up
- does not expire the content (even after inactivity), as we want content here to last
If we want to try to have the embedded audio player enabled for Puzzling, I think the right way to start would be to make a specific feature-request post here on our local Meta, and demonstrate that there is community support for the idea. If we have that support, we can proceed from there.