Would molybdenyl chloride or tungstenyl chloride work as an alternative to chromyl chloride test?
In the chromyl chloride test, chromyl chloride is not a reagent that can be added to the sample to detect any chloride present. It is made in situ through a reaction of chloride containing sample, potassium dichromate and sulfuric acid.

$$\ce{NaCl + K2Cr2O7 + 3H2SO4 -&gt; 2Na2SO4 + K2SO4 + 2CrO2Cl2 ^ (red fumes) + 3H2O}$$

Now in order to make molybdenum chloride work like chromyl chloride, you have to actually make molybdenyl chloride. For this, you need potassium dimolybdate. But it is quite unstable and is prone to decomposition. From here:


  Potassium Dimolybdate, $\ce{K2Mo2O7}$, is formed by fusion of suitable quantities of $\ce{K2CO3}$ and $\ce{MoO3}$, but is unstable; the aqueous extract of the fused mass yields large crystals which immediately begin to decompose. 


Making tungstyl chloride is not feasible because in order to make it, you need potassium ditungstate which is costly, rarely used reagent and can be hardly found in chemical vendor or supplies. So, this method is out of scope and reach. 



Potassium ditungstate or paratungstate has a unique formula that does not follow the general formula of potassium dichromate or potassium molybdate. Its formula is given as $\ce{5K2O.12WO3.11H2O}$ or $\ce{3K2O.7WO3.6H2O}$