The more idiomatic preposition that goes with the job market is usually "in".
The demand for skilled electricians in the job market can vary from country to country.
But I definitely see where you are coming from. I gather you are drawing a parallel between merchandise "on the market" and a person or a form of labor "on the job market".
It is true that we often talk about a product/commodity being "on the market", while the trader of the commodity is "in the market".
How long has this house been on the market?
But as with any other language, English has exceptions to almost every rule and idiomatic usage. Prepositions are tricky. You can also talk about commodities being in the market. This usage is less common but not uncommon:
"It's one of the biggest scandals in my memory," Richard Feigen, an Old Master art dealer in New York, told Bloomberg after last year's revelations about forgeries in the market. (NPR)
It'd make sense your intuition tells you "on the job market" might work. And it does, though not so popular as "in the job market", as this Google Ngram chart shows:
Examples:
Anyone hearing that would assume he was working on a startup, and certainly not on the job market. (source)
In the US market at least, I think it's better for students to show a portfolio of skills on the job market. (source, from Catherine Eckel)
with the over-supply of university degrees on the job market, employers have increasingly sought post-secondary graduates to fill job vacancies in middle-skills jobs that were previously performed by employees with a high school diploma (source)
To sum, your sentence could go either way, although "in the job market" is certainly more common than "on the job market".
In the job market
is colloquial US English. Perhaps you're thinking of "on the job," i.e., at work.