(Yes, I know that recent research shows it was more likely to be humans, not rats, but I’m talking about in the past.)
So when was the connection made?
Paul-Louis Simond, in 1898. At least for the fleas. I’m not sure when they knew that the rats were a vector.
I believe that the correlation of rats and plagues has been around for a very long time. Mass epidemics generally led to the breakdown of services, which led to bodies and garbage piled up everywhere (moreso than normal), which led to swarms of rats everywhere (moreso than normal).