The question: are ammonium cyanate and urea constitutional isomers. My thoughts are no since IUPAC's definition constitutional isomers as
"Isomerism between structures differing in constitution and described by different line formulae e.g. CH3OCH3 and CH3CH2OH." And the definition of constitution as "The description of the identity and connectivity (and corresponding bond multiplicities) of the atoms in a molecular entity (omitting any distinction arising from their spatial arrangement)." Since ionic compounds are not molecular entities then it would never be correct to include an ionic compound as a constitutional isomer.
Any other thoughts?
If you are thinking of ionic solids or ionic compounds in solution, where these compounds have separated into an anion and cation, you are likely correct that the answer is no. On the other hand, ionic compounds can exist (if not easily) in the gas phase, where they could retain their chemical structures as intact molecules.