You might try something like 100% wool felt cloth. Wool will absorb 30% its weight in moisture which is a good wicking percentage. You can get good felt in a variety of thicknesses as well. Felt cloth is made of a lot of different materials these days. Some is even made from 100% recycled plastic bottles - I have no idea how well that would wick the aromatic oils.
If you wanted to create something that was a bit more decorative, you could use a variety of wool felt colors and do a simple design by combining two layers of different colors and cutting holes in the top layer so the bottom color shows through. You could probably find instructions for making felt cloth dolls (like gingerbread men or nut crackers) some of which are stuffed - which would give you the option to stuff it with something like dried lavender which is often used in sachets.
You might consider wet or needle felting where fibers are manipulated into various shapes. There is no law saying that car fresheners need to be flat.
How long the scent would last would probably have more to do with what aromatic oils you use than with the material to which it is applied.
Hope that is of some help to you!!