Nitric acid vs. chloric acid strength
The two ions you are comparing to each other are very different. In $\ce{ClO3-}$ there are two more electrons than in $\ce{NO3-}$. This is also the reason why the former pyramidalizes, while the latter does not.

In nitrate the negative charge can be delocalised over the whole molecule, since there is one molecular orbital with $\pi$ symmetry. One basic requirement for these orbitals is, that they are antisymmetric with respect to a mirror plane, that includes the bonding axis. 

This MO scheme would also be thinkable for the chlorate ion, but here you have two more electrons, so you have to fill an antibonding orbital. The charge can no longer be delocalised as well in this molecule. The conclusion is, that the symmetry of the whole molecule is lowered and the $\pi$ orbitals vanish. A lone pair is always more stable in an orbital with high $\ce{s}$ contribution, which leaves only the $\ce{p}$ orbitals of chlorine for bonding with oxygen.

Resonance is a model which is necessary to describe bonding situations, that cannot be described by one Lewis structure alone. It is therefore more a rule of thumb than any evidence. $\pi$ bonds (or orbitals) are not necessary for valid resonance structures. They are however necessary for ([negative] hyper-)conjugation.