Why is iron pentacarbonyl known, but iron(III) hexacarbonyl is not?
There are several things which are needed to be known to be able to deal with this question.

Firstly there is the 18 Valence Electron (18VE) rule (description at libretexts.org). The iron pentacarbonyl is an 18VE complex. It is coordinatively saturated. A low valent transition metal complex with strong ligand field ligands is considered to be coordinatively saturated if the metal has 18 valence electrons (noble gas electron count).

An iron atom has eight valence electrons, each carbon monoxide (carbonyl) ligand adds two electrons to the valence count. Thus we have a total of 18 in total.

Each positive charge which the complex has causes it to have one less valence electron, so the cationic complex will have an iron which will have 17 VE, this alone would make it less stable.

But there is another issue which is more important, the bonding of carbon monoxide to transition metals is dominated by the withdrawal of electron density from the metal atom to the carbon monoxide. So the less electron-rich the metal the less able the carbon monoxides are to bind.

The fact that the iron in the +3 oxidation state will make it less able to bind to $\ce{CO}$ molecules. This is the main reason why it will not be possible to make the complex.

If you consider the series of complexes $\ce{[V(CO)6]-}$, $\ce{[Cr(CO)6]}$, and $\ce{[Mn(CO)6]+}$, the strength of the bonding between the $\ce{CO}$ and the metal decreases along the series $\ce{V, Cr}$ and $\ce{Mn}$ ($\ce{Mn}$ has the weakest bonds).

Secondly, we should only use the 18VE rule when the oxidation state of the metal is low and the ligands are strong ligand field ligands. I do not think that the iron(III) is sufficiently low in oxidation state for the chemistry to be dictated by the 18 VE rule. I would predict that if I was to bubble $\ce{CO}$ through a solution of iron(III) in dilute acid that nothing would happen.

However, if I was to make a solution of an iron(II) complex of a dialkyl glyoxime then it would be able to react with $\ce{CO}$. Here the iron is in a lower oxidation state and has more electron density on it.