"Convenience" is a noun, so yes, you can have it. You can have any noun. (If anybody knows of a noun that cannot somehow be "had", please let me know.). Your question should have been written: "Can you say something "has a convenience?" Your sentence does not need to say "a convenience" because "its" serves that purpose.
In want-ads for apartments, property owners used to say "all mod cons" as an abbreviated version of "all modern conveniences". It means the apartment has dishwasher, refrigerator, etc.