Did Tony Stark download a copy of the data of JARVIS/Vision in a hard-drive? Or is the data stored in the system gone forever?
We presume the "data" is destroyed
...since there is no evidence of Tony (or anyone else) mentioning it.
Certainly the Vision is primarily cybernetic but it's doubtful if the data was stored in something as simple as hard drive.
Could it be reconstructed?
Possibly, in Infinity War, Banner emphasises that the Vision is more than the Stone
Bruce Banner: Because you might have a choice. Your mind is made up of a complex construct of overlays. J.A.R.V.I.S., Ultron, Tony, me, the Stone. All of them mixed together. All of them learning from one another.
Wanda Maximoff: You're saying Vision isn't just the stone?
Bruce Banner: I'm saying that if we take out the stone, there's still a whole lot of Vision left. Perhaps the best parts.
Other than the stone, the other constituent parts are still "available" (or capable of being reconstructed) in order to make a new Vision.
It wouldn't be the same though since a fundamental part is now missing.