Apparently lead free ammunition is available. I don’t know why you’d be so lazy not to buy it if you’re actually going to eat the meat
The projectile of a piece of ammunition is also used by the feds as an excuse to ban importation if they deem it not suitable for sporting purposes.
There's not a lot that can be done to fix this today, because cartridges were designed with lead projectiles, and guns designed around the cartridges. To get a heavier bullet, it would need to be a substantially bigger(longer) bullet, therefore not working in any gun.
A reasonable proposition is for a new cartridge to be designed around an all copper(or steel, whatever) bullet. It would take a lot of time to get adoption, though, unless the military uses it.
The US military has already moved to steel core, lead free projectiles over a decade ago, specifically because of safety concerns about lead contamination. However, steel core ammunition is explicitly illegal in the US, because it's considered "armor piercing".
That's... not really how that works. The US has tens of thousands of laws related to guns, and they intersect in different ways in different jurisdictions. Federally, though, at any rate, AP ammunition is only illegal for /handguns/. For rifles, there is no such legal restriction and steel core rifle rounds are commonplace. Most hunting is done with shotguns and rifles, you can easily find steel pellet shotgun shells.
The bigger issues is 1) Increased costs 2) Different ballistic performance.