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Find a custom sign and banner printing shop (easy to find even in small towns anymore). Most have facilities to create a full color transfer of any artwork you provide, which they can scan and blow up to size and print on weatherproof plasticized fabric. My brother owns a design company that creates promotional campaigns for businesses and non-profits and he has banners and waterproof fabric signs made like this all the time. Then you can take a plain tire cover and glue the transferred print on or take it to a shoe repair or luggage repair shop and have them sew it on. Here's one site where you can order a 2' x 4' custom laminated banner for under $25. You upload your own artwork or free clipart. Just make sure you don't use anything copyright protected. There are lots of such businesses on line.
Since you would only need about a 2" x 2' design area to cover most of a spare tire cover, you could get the 2' x 4' banner with two copies of the design on it side by side to cut apart and create two covers.
I played around on the site and quickly was able to make a two image design. Try it. Under their "clip art" choose "people" and then "stick man" images. There is a kayaker design like the ones they use in the Olympics. I "pasted" two of these side by side on the banner blank and enlarged them (drag the frame arrows) until they were a good size then chose bright blue color for them. I added text ("Paddle On!") above them with the text tool and changed the letters to red. Then I saved it -- if I had hit "buy now" the order would have been on the way for under $30 with shipping.
You could also upload a different clip art -- there are hundreds of nice looking free kayak themed designs on free clip art sites. I've often found and used them for newsletters and to illustrate articles.
If you've got young kids they would probably have fun creating one of these banner designs.