Translation:We can have ham or chicken with the fries.
There's no requirement for consistency. The fries here is specific in the context. For example, if we're talking about the best fries in town, or the fries that I just made. You can have ham or chicken with THE fries.
If the fries were not specific, it would of course have "des" instead. I can have ham or chicken with fries (avec des frites).
Your example will still be "avec des frites".
Since the sentence is not about the chips itself, the definite is not used. If the general topic is explicitly about the chips, then you'd use the definite article, as in "Les frites sont bonnes" or "Les frites vont bien avec du jambon."
Just a side-note here about "on". It seems convenient to translate it as "one", but the use of "on" is very comfortable in French whereas in English, saying "one" feels a bit stiff. Remember that "on" will often be taken as "we" in the generic sense, and it may also be used in the sense of a generic "you" or a generic "they" if it fits the context.