TL;DR:
- Gas phase reaction (the fastest way to react two substances) between $\ce{HI}$ and $\ce{O2}$ is rather slow and often requires a catalyst.
- Low amounts of dissolved oxygen in the solution will reduce the rate of the reaction to negligible amounts.
- $\ce{HI(aq)}$ is commercially available and is expected to be stable with a long shelf life.
The details
Although, according to various sources, including Wikipedia, $\ce{HI}$ does react with $\ce{O2}$ to form $\ce{I2}$ and $\ce{H2O}$, I am not able to find sources that claim such a reaction in aqueous state; however, several studies have been conducted using catalysts. For example, the reference$^1$ suggests the following reaction mechanism for the reaction between $\ce{HI}$ and $\ce{O2}$:

where $S$ is a surface site. The same article does mention a concurrent slow reaction between $\ce{HI}$ and $\ce{O2}$:

Do note that the reaction has been suggested as being rather slow. Thus, in the absence of a catalyst, the reaction between $\ce{HI}$ and $\ce{O2}$ is not very fast. This study considers very large amounts of oxygen, while oxygen only feebly dissolves in water. The reference-suggested (important) transition state for the reaction between $\ce{HI}$ and $\ce{O2}$ is:

In aqueous state, such a transition state wouldn't be possible since the ions are dissociated and solvated.
Further, since Merck and other commercial chemical manufacturers provide the hydrogen iodide solution, I expect it to be stable with a long shelf life.
References
- Shum, L.G.S. and Benson, S.W. (1983), The oxidation of HI at low temperatures and the heat of formation of HO2. Int. J. Chem. Kinet., 15: 323-339. https://doi.org/10.1002/kin.550150403