A few months ago I lost my Pokemon X and Pokemon Y cartridges and so I bought a new Pokemon y cartridge with no saves on it so I was wondering Can I transfer the save data that I HAD lost in my cartridge that the 3ds has into the new Pokemon Y game to get all my progress back?Progress like my pokemon,items,& story completion?
No, Pokemon X/Y saves to the cartridge, not the system. It is possible to transfer a save from the cartridge to the system IF you have the digital download version, but this does not work the other way (digital to cartridge). And even then you'd need the cartridge to transfer to the digital version.
But since you still have the system, if you have any Pokemon in the Pokebank storage app, you can withdraw them onto the new game. But I'm afraid all the items and story will require playing through the game again.
If, by some chance, you happen to find your lost game, there is still no official way to transfer saves between cartridges. You could transfer Pokemon through the Pokebank. And if you have a second 3DS available (either yours, a sibling's, or a friend's), you could painstakingly transfer one (non-key-)item at a time by trading Pokemon back and forth between your cartridges. I wouldn't recomend using this to transfer anything but the most rare/valuable items like Master Balls, Mega Stones, etc.