Does the compound KFO₃ exist? As the compound potassium chlorate exists, is there something as $\ce{KFO3}$?
All the other halogenates are mentioned somewhere on the web but what about this?
If no, then why?
It has been already mentioned in the comments and the other answer, that fluorine is more electronegative than oxygen. Therefore it is highly unlikely that this compound exists.

To illustrate the problem at hand, take a look at similar, known compound with a fluorine oxygen bond: hypofluorous acid, $\ce{HFO}$, which is actually better described as hydroxylfluoride, $\ce{HOF}$. According to Holleman and Wiberg's[1,2] textbook the synthesis of this compound is already difficult. Low pressure fluorine gas is passed over ice at about $-40~^\circ\mathrm{C}$. Obtained is a mixture of $\ce{HF}$, $\ce{O2}$, $\ce{H2O2}$, $\ce{H2O}$ and also $\ce{HOF}$. Fractional freezing until about $-183~^\circ\mathrm{C}$ leads to a white substance, that turns pale yellow at about $-117~^\circ\mathrm{C}$, and boils at about $10-20~^\circ\mathrm{C}$. It decomposes via a radical pathway at about $25~^\circ\mathrm{C}$ and $100~\mathrm{mbar}$ with a half life of about 30 minutes to oxygen and hydrogen fluoride: $$\ce{2HOF -&gt; 2HF + O2}.$$

The electronic structure of the molecule is most likely $$\ce{HO^{\delta+}\bond{-}F^{\delta-}}.$$

Salts, $\ce{M^{+}OF^{-}}$, that are derived from this compound are not known. Covalent compounds, however, are known, i.e. $\ce{O3ClOF}$ or $\ce{O2NOF}$.

It is easy to imagine, that a compound with an even higher oxygen load cannot be stable. Therefore we can conclude that a halogenate like $\ce{HFO3}$, let alone its salt $\ce{M^{+}[FO3]^{-}}$, cannot exist.

Notes and References  


Unfortunately I only have access to the German version of this textbook. I tried to excerpt and translate the important parts to the best of my knowledge.
Holleman, A., Wiberg, N., Wiberg, E., et al. (2008). Lehrbuch der Anorganischen Chemie. Berlin, Boston: De Gruyter. Kapitel XII 4.2. Sauerstoffsäure des Fluors. S. 465.  (in german)
Translation of the 101st edition available as Egon Wiberg and Nils Wiberg (2001) Holleman, Wiberg. Inorganic Chemistry. Academic Press.