That is an interesting question. It is not a thought that has occurred to me before.
In principle, if you can have oscillating chemical reactions, there is no reason to doubt that some of them might be a basis for an alternating flow of current, but I cannot help doubting that they would be useful. For one thing, all the chemical oscillations that I can think of offhand oscillate comparatively slowly, and I cannot think of a serious need for a chemical battery that produces slow AC, and especially not if you cannot provide the current at a defined or controllable frequency. The fact that I cannot does not prove anything of course, and if you have any ideas...
Secondly, the best-known oscillating chemical reactions are not particularly energetic. This is hardly surprising; really energetic reactions tend to go one way, fast! So any such battery is not likely to supply exciting quantities of electric power. The big interest in batteries nowadays is to produce safe, rechargeable, large capacity batteries; come up with something like that, and your fortune is made.
Thirdly, nowadays there are so many cheap, efficient, effective, and convenient mechanical and electronic means of conversion between DC and AC electric power, that even if we could identify such a purely chemical battery, it is hard to know who might prefer it to readily available alternatives.
Still, as I said, quite an interesting idea.
Cheers,
Jon