Ozone is not mentioned in the synthetic techniques reported by Wikipedia, citing Greenwood and Earnshaw[1].
Summary:
Orthoperiodate ion, $\ce{IO6^{5-}}$, is obtained by alkaline oxidation of iodate either electrolytically on a lead dioxide anode or chemically with chlorine. The product may then be acidi-fied to $\ce{H5IO6}$. Alkaline conditions favor formation of the oxygen-rich orthoperiodate ion; note the inclusion of hydroxide ions in $\ce{IO3^- + \color{blue}{3 OH^-}->H3IO6^{2-} +2e^-}$$\ce{IO3^- + \color{blue}{4 OH^-}->H2IO6^{3-} + H2O +2e^-}$ (the periodate ion is listed partially protonated, as it appears at pH 14 in the Pourbaix diagram from [2]).
Metaperiodic acid, $\ce{HIO4}$, may be obtained by dehydration of the ortho acid. This can be accomplished by heating to 100°C (but below 150°C) or by treatment with dilute nitric acid.
Reference
1. Greenwood, N. N.; Earnshaw, A (1997). Chemistry of the elements (2nd ed.). Butterworth-Heinemann. p. 872. doi:10.1016/C2009-0-30414-6. ISBN 978-0-7506-3365-9.
2. Voutchkova, Denitza & Ernstsen, Vibeke & Kristiansen, Søren & Hansen, Birgitte. (2017). "Iodine in major Danish aquifers." Environmental Earth Sciences 76. Doi: 10.1007/s12665-017-6775-6.