Yes a Connecticut Home Owner may install their own septic tank or septic system on their own property.
You must however get a Permit and a Percolation test done of the soil if there were no previous septic system installed.
If you're replacing the septic system with a new one then usually no new percolation test will need done but some places may require it again.
You'll need to get the permit to install a new septic system but you are allowed to install your own septic system if you're the homeowner or landowner.