Console developers need to use the APIs that are laid out by the system's creator. Nintendo has their own graphics API for the last three generations, called GX, although it is rumored to be somewhat like OpenGL. A few days ago, Nintendo's logo appeared on the Khronos Group's website as a Contributor Member. This leads sites like The Register to speculate that Nintendo “pledges allegiance to the Vulkan (API)”.
I wouldn't be so hasty.
There are many reasons why a company would want to become a member of the Khronos Group. Microsoft, for instance, decided that the small, $15,000 USD/year membership fee was worth it to influence the future of WebGL. Nintendo, at least currently, does not make their own web browser, they license NetFront from Access Co. Ltd., but that could change (just like their original choice of Opera Mini did). Even with a licensed browser, they might want to discuss and vote on the specifics. But yes, WebGL is unlikely to be on their minds, let alone a driving reason, especially since they are not involved with the W3C. Another unlikely option is OpenCL, especially if they get into cloud services, but I can't see them caring enough about the API to do anything more than blindly use it.
Vulkan is, in fact, most likely what Nintendo is interested in, but that also doesn't mean that they will support it. The membership fee is quite low for a company like Nintendo, and, even if they don't use the API, their input could benefit them, especially since they rely upon third parties for graphics processors. Pushing for additions to Vulkan could force GPU vendors to adopt it, so it will be available for their own APIs, and so forth. There might even be some learning, up to the limits of the Khronos Group's confidentiality requirements.
Or, of course, Nintendo could adopt the Vulkan API to some extent. We'll see. Either way, the gaming company is beginning to open up with industry bodies. This could be positive.