Yes, you do. The FDA passed a law not too long ago that changed how all contact lenses were viewed.
Before, colored contact lenses were looked upon as simply recreational but since there were companies and individuals selling lower quality, people who bought them ran the risk of getting eye infections from unclean or improper, cheap or otherwise just bad contacts.
So they passed the law, which you can view here, which makes all contact lenses, whether they be vision correction or not, as medical devices, which require a Doctor or a contact lens professional to prescribe to someone.
If you plan on buying your red contacts online, then you will need to prove to that online company that you do have a prescription by going through the proper channels.
Fortunately, it’s not difficult to get a prescription, you can visit your local Doctor or Eye Care specialist and let them know you are wanting to buy a red contact lens and they should have no problems giving you a prescription on the spot, as long as you don’t have any medical reason that you shouldn’t be wearing them.