Does this 'Belousov-Zhabotinsky Reaction' kit contain the correct reagents?
No, it doesn't, at least not strictly.
According to Wikipedia (emphasis added):

One of the most common variations on this reaction uses malonic acid $(\ce{CH2(CO2H)2})$ as the acid and potassium bromate $(\ce{KBrO3})$ as the source of bromine. The overall equation is:
$$ \ce{3CH2(CO2H)2 + 4BrO3− → 4Br− + 9CO2 + 6H2O} $$
[ $\ldots$ ]
Many variants of the reaction exist. The only key chemical is the bromate oxidizer.

The kit doesn't contain any bromide or bromate, so by definition it will not allow demonstration of a Belousov–Zhabotinsky reaction.

However: It does appear that the kit may allow demonstration of an iodine clock reaction:

Hydrogen peroxide variation
This reaction starts from a solution of hydrogen peroxide with sulfuric acid. To this is added a solution containing potassium iodide, sodium thiosulfate, and starch.

The ingredients are not a perfect match, and actually appear to be something of a mishmash between the iodine clock reaction and the Belousov–Zhabotinsky reaction, but they're in the right ballpark. Metal ions and their oxides are well known as catalysts for hydrogen peroxide decomposition, and &quot;Sulfamic acid&quot; could be a typo that was supposed to be &quot;sulfuric acid&quot;.