Is it illegal to make a door wreath with Crimson Tide or War Eagle ribbon I purchased with?

If you don't make anything of a scandalous nature or that paints them in a bad light AND you purchased the starting material from an authorized dealer AND you make it clear from your hang tags that the wreath came from your company and not the football team, - then you should be OK.

No. You made a legal purchase. You may use the ribbon any way you want.

If you are purchasing items with some intellectual property claim upon them (patent, trademark, copyright), you can do whatever you want with the items. However, if you want to sell your wreaths with these features to the general public, you may need permission from the IP owner.

Making a door wreath using trademarked ribbon (presuming you purchased the ribbon from an authorized seller) for your own personal use would be legal. The problem could arise where even after you purchased the ribbon, you then create a NEW product (wreath USING registered trademarked ribbon without trademark owner's permission) and start selling your new product. Under U.S. Federal Law, owners/rights holders of federally registered trademarks can sue others who use those marks without permission, when the use in question causes a "likelihood of confusion" to actual or potential customers.