The Model III uses 4116 chips. These are still generally available although you need to be careful of the source. Fake chips (or chips that will fail) are very much a risk, so it is important to source them from someone who has run them through a RAM tester. There are people who sell based on how much they have on hand, and you can always try your luck on eBay.
If you are willing to not have a TRS-80 keyboard and would accept something more modern, the folks over at Plaidvest Software have noted that you can install a Wireless USB Jelly Comb KS037 into a Model III (once you take out the existing keyboard) with a NEWKEY/80. They also have a Thingverse to manufacture a keyboard support if you glue a washer to the underside of the keyboard.
If you need a replacement badges for your Model III, NewSoft sells them for a very reasonable price.
Power Connector Assembly
If you would like to use a standard computer plug instead of the one permanently fastened to your Model III, Newsoft sells a repalcement power connectory assembly. Note: Will require you to open your Model III and do some disassembly and re-wiring.
If you would like to repaint your Model III, you might want to read the How to Paint your Model III or 4 guide by Patrick Bureau, downloadable Here.
For hard drive and floppy drive replacements, visit: