Chlorine can be found as the following oxoanions $\ce{ClO–} , \ce{ClO2-}, \ce{ClO3-}, \ce{ClO4-}$ in many chemical stockrooms. However, iodine is never found as $\ce{IO4-}$. Explain this observation.

I didn't really find any reason as to why periodate cannot exist, unlike anions such as flourate where it is limited by flourine's small size. After researching this further, I found that the periodate anion actually exists in compounds such as sodium periodate. Even though it has another form $\ce{IO6^{5-}},$ the periodate anion actually dominates its equilibrium. Thus, I was wondering what the intended/likely answer was?

Yes, $\ce{IO4-}$ does exist and it is called metaperiodate anion. It can combine with a number of counter ions to form periodates, which may also be regarded as the salts of periodic acid.

Metaperiodates are typically prepared by the dehydration of sodium hydrogen periodate with nitric acid.

$$\ce{Na3H2IO6 + 2 HNO3 → NaIO4 + 2 NaNO3 + 2 H2O}$$

They can also be generated directly from iodates by treatment with other strong oxidizing agents such as hypochlorites:

$$\ce{NaIO3 + NaOCl → NaIO4 + NaCl}$$

Metaperiodates adopt a distorted tetrahedral geometry with an average $\ce{I–O}$ distance of 1.78 Å

The ortho- and metaperiodate forms exist in equilibrium.

$\ce{H4IO6− ⇌ IO4- + 2H2O, K = 29}$

For this reason orthoperiodate is sometimes referred to as the dihydrate of metaperiodate, written $\ce{IO4^-·2H2O}$

One example of salt having metaperiodate ion is sodium periodate(@Gert). Its preparation is mentioned above. In fact, all other alkali metals and alkaline earth metals forms metaperiodate salts. Some examples include potassium periodate, rubidium periodate, calcium periodate etc.