My understanding is that organized furry random came about from the underground comics scene of the 70s, but that kind of feels like saying that oral sex began with Deep Throat.
Given that it doesn't seem that hard to make anthropomorphic costumes, and that people have been horny for forever, do we have records of people dressing like animals and engaging in sexual activity?
Believe it or not, we've gotten questions like this before. u/commiespaceinvader has a great answer to When did furries (people who dress up in animal costumes) become a thing?
One of the key points with furry fandom is that it is important to distinguish between the activity and the identity. Not all furries dress up like animals to engage in sexual activity, and not all people that put on animal costumes and engage in sexual activity identify as furries. Historical examples tend to fall under the latter: just because a noble dressed in a lion costume for a masquerade ball and had sex with a courtesan doesn't necessarily make them a furry, or even a spiritual forebear of such.