This is quite possible and very frequently seen in graphics. This is commonly referred to as a "look up texture" (LUT) when used as an intermediary in calculations.

You can use all kinds of parameters besides position as your texture coordinates, too, such as other intermediary values. There are no rules about what the coordinates used in a texture access are supposed to represent. They could be normalized angles or intensity values or noise input or whatever you need.

Whether you need a 2D texture, a 3D texture, or a cube map is not something anyone could say without you telling us what you're actually trying to do. :)

Sounds to me that you're after a 3D texture. A cube map consists of the 6 faces of a cube, it's hollow - there's nothing inside it. A 3D texture is the name for what you're after, a solid cube (or cuboid) of texture.