It depends on the grenades you wanted.
Just about anyone can buy a smoke grenade, and for civilians they're the best kind. They're useful for a lot of things. Some boaters carry them--if they're in trouble on the water, they'll pop a floating orange smoke grenade and throw it in the water, where people can see it from a long way off. You could also use them for special effects--if you wanted your high school football team to come out in dramatic fashion, pop a smoke grenade and have the players hold their breath as they run through the smoke.
Destructive grenades--frags, incendiaries, white phosphorus--are a totally different story. They are classed as Destructive Devices. To get one you need to file a form, get approval from the sheriff, get fingerprinted, go through a background check and pay a $200 tax for every grenade you want. It's not worth it.
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The answer to this depends on what kind of grenade you want. There are many kinds.
If you want a fragmentation grenade, the answer is probably not. Technically you can get one, but you need the permission of your local sheriff and you need to do a BATF Destructive Devices transfer, which costs $200 per grenade. When you go to the sheriff, he will probably look at you, laugh, and ask, "now, what do you want THAT for?"
If you want a smoke grenade, that's a different story. You can buy those online, and paintball players DO. These are lots of fun. Really.