Tanning is the process of treating skins of animals to produce leather. Once the skin is turned into leather, then it is sometimes dyed.
The tanning process is indeed smelly, but unless you're planning on starting your project with a dead cow/goat/deer, you shouldn't need to do any tanning.
If you are after
very thin leather i.e. less than 1mm, then the sort that bookbinders use may well suit your needs. I use it for things like covering scabbards. I've bought offcuts in all sorts of colours e.g. brown, black, green, blue, red, purple. You can even paint it with acrylic, if you want to change the colour.
Thin vegtan leather is great for making stuff too, like straps, belts, shoes, cap peaks.
Tony Barton on the OSW forum is an expert at making 1/6 bits in real leather - you could try PMing him on there for advice on where to source thin leather, if you struggle to find it.