I wouldn't say ammo is easier to regulate at all. The average gun owner keeps several boxes of handgun ammo and 500-1000 rounds of rifle ammo on hand to start so none of that ammo will be caught by your regulation.
If you outright banned the new purchase of ammo it would take years before the effects are really noticed, especially considering the rush to buy up every bullet on the market before the ban went into effect. Afterwords you'd see gun owners driving to a state without an ammo ban.
If you slapped a max limit on ownership, say you can't have more than 50 rounds of ammo per gun, that wouldn't have stopped almost any of the mass shooters in the last 10 years who almost all fired less than 50 shots on their rampages. Not to mention enforcing that law would be next to impossible because you'd basically need a signed warrant to check a house for illegal amounts of ammo or catch them in transit.
If you put a massive tax on ammo combined with a max purchase limit, let's say 1,000% tax and a limit of 50 rounds per ID card per day, it would be business as usual, just more expensive.
One thing that might be worth exploring would be an ammo purchase registry. Gun owners haaaaaate their names on lists aand fight it tooth an nail on guns, but they don't have the protection of the 2nd amendment for ammo. Who knows it might have a chilling effect. Or just drive more transactions through the gun show/private seller loop holes.