Translation:We did not find anyone who could stay with the dog.
It's used with tense meaning.
If I say "ninguém que possa ficar", It's subjunctive too, but present. I'm still planning to travel, and I'm searching for someone that "could = can" stay with the dog. Or even have already given up traveling, but the time when the person is supposed to stay with the dog hasn't passed yet. So there might yet be a solution.
If I say "ninguém que pudesse ficar", it's past, so I have already tried to find someone, already lost my trip, because there "was" nobody that could (at that time, which is already gone) stay. Here there's no solution at all.
Just curious, is there a difference in Portuguese between "anyone who..." and "anyone that..." In English, "who" is often preferred when referring to a person, and "that" when referring to an object or animal. (Although "that" is also commonly used to refer to a person and some style guides make no distinction.)