What is the type of hybridization of CO₃²⁻?

CO2  is a linear molecule

   O :: C :: O  The bonds to the left are a sigma and a pi bond forming the double bond .. to the right are exactly the same
  ===  this is sp hybridization,  not sp3

CO3(2-)  is a trigonal planar molecule .. flat .. in one plane and the 3 oxygen atoms form a triangle.
   You are correct ... this is sp2 hybridization.  The central atom, C,  has one double bond(containing a sigma bond and a pi bond) and two sigma bonds with NO unshared e- pairs on the central atom (one of the O atoms has unshared pair, but that has nothing to do with the shape .. just those on the central atom)  ====  the central atom is the C and it has 3 bonds which is sp2 hybridized.
 

The figure below shows the resonance structures of CO₃²⁻ ion.

In each resonance structure, the 4 valence orbitals of the central C atom form 1 C=O double bond and 2 C-O single bond.  Each double bond contains 1 σ bond and 1 π bond, while each single bond contains only 1 σ bond only.  Hence, the central C atom has 3 σ bond and 1 π bond. 

Each hybridized orbital forms a σ bond or a lone pair of electrons, while each unhybridized 2p orbital forms a π bond.  Hence, because the central C atom has 3 σ bonds and 1 π bond. the central C atom has 1 unhybridized orbitals and 3 hybridized orbitals, i.e. 3 sp² orbitals.

You are right.  The type of hybridization is sp².