Can a polyatomic ion have no dipole moment?
Yes if all atoms within the molecule are the same, which in reality means no.
If two (or more) different kinds of atoms are bonded you will always experience a polarity of the bond which in turn will lead to a dipole moment.
In practice, these dipole moments can sometimes be so small that they are for the most part considered negligible. Sometimes, however, their effect is great - water is a great example of this.
Polyatomic molecules never really have only one kind of atom in them (usually that is diatomic molecules) and so usually do have a dipole moment. If however, all bonding atoms were the same or overall electron distribution was ‘equal’ then yes, it is possible for there to be no dipole moment.