The web player fails to show button textures. Images display in the buttons until the web player build compresses the textures and then they no longer display on the buttons.
How can we reproduce it using the example you attached.
Open the attached scene. Potentially reimport the texture to decompress. Launch in the editor. The textures display on the buttons. Then build the web player which compresses the texture and the button textures will appear blank.
I guess the work around will be to not compress the textures (in the import settings) so they can appear on GUI elements.