I don't think there's any ambiguity in your example. You said that the fox is trying to find the rabbit, so it's the fox that finding, not the rabbit. Thus when you say "when it finds one", clearly "it" refers to the fox and "one" to the rabbit. I suppose theoretically "it eats it" could be ambiguous, but a reader would normally understand that the fox eats the rabbit, and not the other way around.
That is, we can often tell who or what a pronoun refers to from our general knowledge of the world. If the parties conform to common stereotypes, the meaning is clear. If not, if the roles are switched, then you need to reword the sentence to make this clear. "The general met the private and he ordered him to return to the barracks." We'd normally assume that the general gave orders to the private and not the other way around. If it really was the other way around, if the private is giving orders to the general, then you need to reword to make that clear.
In general, a reader assumes that the subject remains the subject, the "active party", unless otherwise specified. For example, consider a slight variation of your example. "The policeman was hunting for the criminal, and when he found him, he shot him". I and I think most readers would understand this to mean that the policeman shot the criminal, and not the other way around, even though it is quite plausible to say that the criminal shot the policeman. We assume that the same person, the policeman in this case, remains the subject throughout. If you wanted to say that the criminal shot the policeman, you'd have to specify that. Like, "The policeman was hunting for the criminal, and when he found him, the criminal shot him."
When the pronouns are different, the problem goes away. "Sally was looking for John, and when she found him, he gave her the box." There's no question about whether "he" refers to John or Sally, because we assume that Sally is a "she" and John a "he". If the sentence used names of ambiguous gender, the problem comes back: "Kelly was looking for Tracy ..." And of course it's possible that someone has a name that we normally associate with the opposite gender.
When in doubt, replace the pronouns with nouns. "Sally was looking for Mary because she was supposed to give her the box." It's not clear who was supposed to give the box to whom. So change one of the pronouns to a noun: "Sally was looking for Mary because Sally was supposed to give her the box." Now it's clear.
Sometimes using nouns would lead you to use the same noun many times in one sentence, which can get awkward. It's helpful if there are multiple nouns you can use to identify a party. "Sally was looking for Mary because she was supposed to give the tall girl the box." If in context we have previously identified one of them as "the tall girl", this makes clear which is which without having to repeat one of the names.