First I want to differentiate between the 'solar company' who sells you the system and the 'solar installer' who installed the system.
These can be the same entity, or they can be different entities.
The company (or sole trader) who sells you the system needs to hold their own electrical licence in:
In the following states the company selling you a system does not need its own licence - they can contract out the installation to a sub contractor with the correct electrical licence:
- TAS (with conditions)
If subcontracted to another company, the company who is installing the system also needs an electrical licence.
The individual installing the system needs a personal electrical licence as well as Accreditation to install solar from the Clean Energy Council. Note the CEC accredits people not companies. There are 2 levels of Accreditation: On-grid and off-grid. Your installer needs the latter if you have decided to go off-grid. Also, the design of your solar system must be done (and signed off) by a CEC accredited Designer. Ask for the design document with this signature.
If you want to go with a non-accredited installer, and are not claiming any 'rebates' (AKA STCs) and are not connecting to the grid you can, but I would not recommend it.
There is also a more detailed blog post about the levels of Clean Energy Council accreditation/membership/approval here.
Finally you do not need to be a CEC Approved Solar Retailer to install solar - it is a voluntary scheme.