Can a network covalent solid be amorphous?
The most obvious example is amorphous carbon. As mentioned in a comment, another good example is glass (although, whether it's truly "solid" past the glass transition temperature). 

Really, a "network" simply means "solid with covalent bonding" and of course it's easy to imagine many possible amorphous network solids.

Vulcanized rubber might qualify too, but it would depend on whether there were many non-interlinked macromolecules. Usually macromolecules would be considered molecular solids due to the high number of non-bonded packing interactions. If the amount of cross-linking is high, classifying as a network solid makes sense.