Is it illegal to make your car have no emblems?

No, it is perfectly legal to remove your emblems. They play no bearing in the legality of your vehicle. The emblems put on the car from the manufacturer are just for free advertisement on their part.

You can change pretty much anything you want about the styling of your vehicle provided the vehicle is still safe to operate on the road. About the only thing you cant legally change about the vehicle is its VIN (vehicle identification number) tags, usually located on the dash and firewall (other places as well on newer vehicles)