Can solutions of polar covalent compounds conduct electricity?
Consider for example this lab where polar covalent bonds are said to be strong, weak or non-electrolytes : ELECTROLYTES AND NONELECTROLYTES CHEM 151

The part of the molecule which dissociates has strongly ionic character but the molecule itself is described as being polar covalent. Not not all bonds in the molecular have the same level of covalent character, but most are covalent and it is polar.

There are also other ways for molecules to allow conduction, conjugated systems can do it by a very different process see for example the wiki on Conductive Polymers