Why is the periodate anion not found?
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.