Applying texture without coordinates

Is there any way to apply the texture to an object without specifying texture coordinates?

In fixed-function OpenGL, you can generate texture coordinates by activating texture coordinate generation modes. There are a couple of fixed algorithms (spherical coordinates, reflection coordinates), and there is one that multiplies a vertex component by a 4x4 matrix to generate a texture coordinate.

In shaders, you can use anything you can algorithmically generate.

However, without telling us how you want a texture mapped to the surface, there's no way to know if what you want is possible. There is no glTextureMyObject that does "something"; either explicit texture coordinates must be used or some algorithm must generate them.