To answer your question, NO. You do not have to buy a 'special" Lapua bullet. Lapua makes bullets, so does Sierra, Nosler, Hornady, Barnes , etc...Lapua developed the .338 Lapua magnum by necking down a .416 Rigby cartridge to accept a .338 cal bullet. Hence the name .338 Lapua magnum. Just like Winchester developed the .308 Winchester, but you do not have to buy Winchester brand bullets.
So, any .338 cal (diameter) bullet will work in a .338 Lapua magnum, or .338 win mag, or .338 Remington ultra mag etc.
I have a .338 Lapua. My load is 91 gr Retumbo and a 300 Grain Sierra match king. It's right around 2900 fps with a 27" barrel and groups 7" at 1000 yards. I use Lapua Brand brass, Hornady brass is a bit soft, HSM is OK though. That load is hotish so start closer to 84 gr and work up.
Good luck