Is there an analogue of OpenGL's ImageLoadStore in DirectX 12? I'm trying to port part of an OpenGL program into an existing DirectX 12 software, at some point in their shaders they use OpenGL ImageLoadStore in order to store information in 3D Textures without having to use an FBO.

Is there something similar in the DirectX 12 API? 
Or would I have to render to a custom FrameBuffer?
You should be able to use RWTexture3D.
The hlsl documentation page about RWTexture2D also has sample code that you might find useful.