In Sense and Sensibilia, Aristotle laid out arguments against some of his contemporaries in which he argued against the idea of a finite speed of light. He argued it was an infinite speed. So we can trace back some realization to the speed of light going back to ~300-400 BCE.

Ole Rømer was the first recorded man to measure the speed of light accurate to an order of magnitude in 1676. Here's some info on this.

That's a pretty huge range. Somewhere in it exists the first person who had a proper, rational reason to believe light's speed was finite, but until it could actually be measured/tested, it's not really science. Testability is a cornerstone of good science. As such, I think it's safe to drop the time period in the early 1600s, when testability first became possible.