As already mentioned, Nintendo are free to use OpenGL, Vulkan, SPIR, etc. or derivatives thereof without being members of Khronos, which is what makes them open standards. A company only benefits from being a contributing member of a standards-setting organisation like this if they're concerned about interoperability between their hardware/software and that of other platforms. For example, Nintendo could either add features to a Vulkan-based API for NX, which would then only work on Nintendo platforms, or they could convince the Khronos group to add the features to Vulkan itself, meaning they'd work on any platform which supports Vulkan.
Now, the question is why would Nintendo want this interoperability, and with which of Khronos's APIs?
The obvious answer here would be OpenGL ES, as Nintendo move into mobile app development it would be advantageous for them to use a single API across iOS, Android and their own hardware. Being able to contribute to OpenGL ES (and Vulkan when it supersedes ES) may allow Nintendo to work in features which allow them to make more effective use of both their own hardware and others' when developing cross-platform apps.
Another possibility is WebGL, for much the same reason, as Nintendo may use it as a means to share content across mobile devices and their own hardware.
Another less likely prospect, although one which shouldn't necessarily be dismissed out of hand, is that Nintendo is moving away from custom APIs and plan on using vanilla Vulkan as the graphics API on NX, either as a means to make porting to the console as straight-forward as possible, or to allow them to switch hardware partners in future while still retaining forward-compatibility. If they were to do this, they'd want to make damn sure that Vulkan could do everything they need it to, which would necessitate becoming a contributor to the standard.
