Can anyone install their own avionics?
As has been pointed out, anyone can install their own equipment provided that a suitably qualified engineer is prepared to sign off the installation.
Of course, the installer will require the correct tooling for making off high-density crimp connectors (if fitting almost all Garmin units), will need access to the correct cables (must meet correct flammability requirements etc). The correct test equipment to prove serviceability doen't come cheap - to test a Mode S transponder is giving the right address needs a $15K test set, plus a suitable air data test set to check the altitude is correct - another $10K! All good avionic shops will possess all of this kit, and pay to keep it calibrated.
A reputable Part 21J design company will specify all the suitable materials as part of the modification data package. Typically a minor mod (e.g. Mode S in a single-engine aircraft with no DAPs) will cost around £700-900. Whilst this is a far cry from paying the old £77 fee to the CAA, the data is no longer a two page application to the CAA regional office, but rather more like 25 pages to satisfy all of EASA's rules including continued airworthiness, full wiring diagram package etc.
Please don't knock all the avionic companies for what they charge- it's generally much cheaper than you pay for car maintenance, has greater overheads due to test equipment and approval fees, requires significantly more training (certainly for avionics) and needs a minimum of 3 years to get through the exams before you can sign of the work.