The question asked is "Why do chromyl bromide and chromyl iodide not exist?" To answer this question, one needs to go back in literature as far as 1824, when chromyl chloride was first prepared by Berzelius by distilling a mixture of a chromate, sodium chloride, and sulfuric acid (Berzelius had not believed he made chromyl chloride at the time).[1] Since then, to my knowledge, only chromyl fluoride, chromyl chloride, and chromyl nitrate of the all reported chromyl compounds have been prepared in the pure condition (note that The chromyl group has been reported to form compounds with all the halogens, and with the nitrate, cyanate, and thiocyanate groups [2]. Let's forget about chromyl nitrate, but, my question is why we don't want to know about chromyl fluoride here as well, since it is also a chromyl halide? Keep in mind that chromyl compounds are physically volatile materials, but there are wide differences in both the volatility and the liquid range of the well-characterized chromyl compounds. As liquids, all these compounds are mobile, dark red in color, and miscible with chlorinated solvents.
Since Berzelius' discovery, many works have been done on chromyl chloride. For the chromate, the choice of the compound among many workers was potassium chromate, although some used potassium dicromate. Measurements of the melting point of chromyl chloride have not been established with high accuracy, since a thermometer calibrated only in whole degrees was used at the time. The accepted value is $\pu {-96.5\pm 0.5 ^\circ C}$, which is reported for this constant in 1912.[2,3] A large number of values have been reported for the boiling point of chromyl chloride. These values, corrected to $\pu {760 Torr}$ where necessary by means of the vapor pressure curve (given in reference). The value of $\pu {116.7 ^\circ C}$ has been accepted by reference works.[3]
Although the chromyl chloride test is a qualitative test for chloride compounds, certain chlorides are reported not to yield chromyl chloride on treatment with sulfuric acid and a chromate. These are the chlorides of the noble metals (platinum, gold, silver, mercury) and antimony(III) oxychloride [2].
The preparation of chromyl fluoride has received far less attention. Most researchers interested in chromyl fluoride had used the reaction between a fluoride, a chromate, and sulfuric acid to volatilize chromyl fluoride [2]. For instance, the elusive vapors of a red chromium-fluorine compound were observed upon heating a mixture of fluorspar (the mineral form of $\ce{CaF2}$), chromates and sulfuric acid as early as 1827, yet, for a long time it was considered to be $\ce{CrF6}$.
Chromyl fluoride was prepared from $\ce{CrO3}$ and anhydrous hydrogen fluoride and isolated in pure form for the first time by Engelbrecht and Grosse in 1952.[4] The purified chromyl fluoride was reported as the violet-red crystals, which reach a pressure of $\pu{780 Torr}$ at $\pu{29.6 ^\circ C}$ and melt to an orange-red liquid at $\pu{31.6 ^\circ C}$. The vapor pressure of the compound was also reported as $\pu{885 Torr}$ at the triple point. Pure chromyl fluoride is reported as stable on storage (according to the reports, at least it makes stable, non-volatile triple addition compounds with $\ce{HF}$ and $\ce{KF}$ or $\ce{NaF}$), whereas the impure material prepared by earlier procedures by others was reported not stable and possessed variable physical and chemical properties. However, alike chromyl chloride, fluoride compound is extremely reactive, particularly with any type of oxidizable organic compound (thus, this may be one of the reasons for its instability when its impure).
Albeit the previous attempts to prepare chromyl bromide by reacting chromates with bromides in the presence of sulfuric acid had been failed, the presence of chromyl bromide was reported in 1911 when anhydrous hydrogen bromide in carbon tetrachloride was used as bromide and chromium(VI) oxide or dilute chromyl chloride in carbon tetrachloride as the chromate. Further, actions of acetyl bromide on chromium(VI) oxide in glacial acetic acid or chromyl chloride in carbon tetrachloride were also resulted a deep transient purple color resembling that of permanganate, which has been ascribed to the presence of chromyl bromide.[5] This intense permanganate-red coloration appeared and then faded rapidly to red-brown solution, further studies of which have shown that it was decomposed to $\ce{Br2}$.
The preparation of chromyl bromide (as an unstable compound) is supported by four different reactions, all of which have yielded the solutions of characteristic permanganate-red color:
$$\ce{CrO3 + 2HBr <=> CrO2Br2 + H2O}$$
The author reported that the solution has immediately assumed a permanganate-red color, which slowly faded to orange-red color.
$$\ce{CrO2Cl2 + 2HBr <=> CrO2Br2 + 2HCl}$$
The addition of hydrogen bromide solution to very dilute chromyl chloride solution (both in $\ce{CCl4}$) has produced an intense permanganate-red coloration, which has gradually faded to orange-red color.
$$\ce{(HO)2CrO2 + 2CH3COBr <=> CrO2Br2 + 2CH3CO2H}$$
Here, $\ce{(HO)2Cr2O2}$) was prepared in situ by adding glacial acetic acid to $\ce{CrO3}$). Thus, the addition of acetyl bromide solution to $\ce{CrO3}$) solution with trace of glacial acetic acid (all in $\ce{CCl4}$) has undergone a reaction, which may be described by above equation, producing an intense permanganate-red coloration, which has gradually faded to orange-red color.
$$\ce{CrO2Cl2 + 2CH3COBr <=> CrO2Br2 + 2CH3COCl}$$
The addition of one drop of acetyl bromide solution to extremely dilute chromyl chloride solution (both in $\ce{CCl4}$) has produced the permanganate-red coloration, which has vanished in the course of five minutes. This color was later used as an indicator (the presence of trace amount of chromyl chloride) in studying the action of chromyl chloride on phosphorus halides.[6] Forbes and Anderson have reproduced Fry's results in 1943, during which the similar colors were formed from the action of boron tribromide, silicon tetrabromide, and stannic bromide on chromyl chloride in carbon tetrachloride.[7] They suggested the decomposition of $\ce{CrO2Br2}$ is due to the reaction: $\ce{CrO2Br2 -> CrO2 + Br2}$.
The purple color was ascribed to addition compounds by Zellner, who finally isolated red-brown needles of chromyl bromide (though contaminated with $7.5\%$ of chromyl chloride), by the action of a tenfold excess of liquid hydrogen bromide on chromyl chloride at $\pu{-60 ^\circ C}$ to $\pu{-80 ^\circ C}$. Chromyl bromide decomposes at room temperature and could not be further purified.[8]
It was reported that, although there were no direct evidence to prove the results, chromyl iodide might be formed by the action of acetyl iodide on $\ce{CrO3}$ in the presence of glacial acetic acid, according to indirect evidence [5,7]. Others using various combinations of mixtures including potassium iodide and sulfuric acid, and various techniques, have obtained no evidence of the existence of the compound. It is therefore presumed not to exist.
References
J. J. Berzelius, Annalen der Physik und Chemie, 1824, 1,
1-48.
The Chemistry Of Chromyl Compounds: Winslow H. Hartford, and Marc
Darrin, Chem. Rev., 1958, 58(1), 1–61
(https://pubs.acs.org/doi/abs/10.1021/cr50019a001).
E. Moles and L. Gomez, Zeitschrift fur Physikalische Chemie,
1912, 80(5), 513-530.
Pure Chromyl Fluoride: Alfred Engelbrecht, and Aristid V. Grosse,
J. Am. Chem. Soc., 1952, 74(21), 5262–5264 (https://pubs.acs.org/doi/10.1021/ja01141a007).
The Chemistry Of Chromyl Compounds: Harry Shipley Fry, J. Am. Chem.
Soc., 1911, 33(5), 697–703
(https://pubs.acs.org/doi/10.1021/ja02218a006).
Reactions in non-aqueous solvents I. The action of Chromyl chloride
upon phosphorus trihalides: Harry Shipley Fry, and Joseph L.
Donnelly, J. Am. Chem. Soc., 1916, 38(10), 1923–1928
(https://pubs.acs.org/doi/10.1021/ja02267a002).
Cyanates and Thiocyanates of Germanium, Phosphorus, Sulfur and
Chromium: George S. Forbes, and Herbert H. Anderson, J. Am. Chem.
Soc., 1943, 65(12), 2271–2274
(https://pubs.acs.org/doi/10.1021/ja01252a003).
Versuche zur Darstellung von Chromylbromid: H. Zellner, and O.-O.
Bad Ischl, Monatshefte Fur Chemie, 1949, 80(3), 317-329
(https://link.springer.com/journal/706/80/3/page/1).