Hola Dawn

In this specific sentence, it wouldn't be natural to use the pronoun because "encontrarse algo" generally means that you find something as a surprise/ by chance.

See this lesson here for different uses of encontrar/encontrarse. 

In this case it is obvious that the person is actively looking for the keys: "cuando encuentre las llaves". This doesn't mean that it's somebody else's keys. We actually use the definite article (el, la, los, las) to refer to things that are obviously ours - no need to use anything that expresses the possession. 

I hope it clarifies it.

Saludos