Yes, sort of, but not how he wants to
Yes, you can use a Reaction to trigger a Contingency. That's fine and pretty straightforward. You have to set up the Reaction ahead of time though, as usual: either by threatening a creature's space and it causing you to gain an Attack of Opportunity Reaction, or by Readying an action ahead of time (every round) causing you to gain a Reaction opportunity.
Because blinking isn't an attack that can take advantage of an AoO, that means he's limited to Readying “blink very hard when attacked” to gain a Reaction. That sounds like a terrible waste of his Reaction and Action every round that he isn't attacked, but it's possible to do. Terrible, but possible.
What he can't do is gain a Reaction option from nowhere — just like Bonus Actions, he can take up to one Reaction, but that doesn't mean he can choose to make a Reaction just because he wants to. A PC has to have an option that they're already allowed to take (like the one created by Readying), and that option (when used), counts as their one Reaction for the turn. You can't just make a Reaction because you want to.
That's what he's trying to do: get a free Reaction opportunity without having any ability allowing him to take a Reaction then.
What about things that wouldn't take an action? Blinking isn't an action!
But isn't the trigger trivial? Why does it need a whole Reaction? You can just talk whenever, after all, and blinking is something we do all the time already.
The problem is that as a reaction to an attack it's not just doing something anymore, it's doing something with specific timing in response to an attack. To pull that off you have to get the timing right, or it doesn't work. There's a mechanic for this conveniently built into the game, that dictates how such critical-timing activities can be done in reaction to a situation: the Reaction action type, and Readying to prepare one.
So although blinking very hard is inconsequentially trivial on its own, the consequence of timing it wrong in response to an attack is such that it can't be done for free anymore, and the most appropriate model for such an endeavour is the Reaction mechanics, including Readying to acquire a Reaction opportunity.
As a last note, to cover any objection to the assertion I make above that you have to get the timing right or it won't work: if you don't want the precision of Readying for your Reaction, then you're OK with just doing it whenever. And that's what your own turn is for.