Anybody can do the installation, but the appropriate person needs to inspect and sign it off.
However, "customer assisted" work is rare in practice, due to the usual commercial issues referred to by e.g. wigglyamp above.
In any case, anybody can legally dismantle a plane or any part of it. It's only putting it back together that is a problem. This pretty obvious concession can save a great deal of money, in the right circumstances.
The really basic problem with avionics is that very few of the installers are electronics engineers. They are largely wiremen, following wiring diagrams in the installation manuals. Most common avionics we see around are just very common and the wiring is well known. The manufacturers also offer a degree of assistance to the installer. But if something just doesn't work then often it is abandoned because nobody is able to get to the bottom of it. Occassionally, the only way to find out why something isn't talking to something else is to get a protocol analyser and virtually nobody is going to do that.