Yep, .223 Remington. Cheap ammo is plentiful, even if you avoid steel cased ammo, and it will shoot fine out to 300 yards (farther with the right barrel twist and bullet combinations).
There are lots of other choices, but for "cheap ammo" + 100 yards, .223 Remington is your boy. And if you ever decide to take up reloading, .223 Remington is not a bad cartridge to start with.
A .22 LR will shoot 100 yards (my dad used to shoot small bore silhouette out to 100 yards with a 10" .22 LR barrel), but it has what's known as a "rainbow trajectory" at that point.