Can functional groups exist by themselves
A functional group on its own essentially has at least one non-bonded electron, thus making it a free radical, this makes them highly reactive molecules, making them react with other substances (including themselves) and can thus only exist in very low concentrations in inert medium/vacuum. 

However, there are molecules such as $\ce{NO}$, $\ce{NO2}$ which do have an odd electron in their molecular state, can exist in reasonably large quantities, and can be functional groups in organic chemistry, these are in some sense functional groups that exist as molecules.