What kind of bond exist between carbon and oxygen in C0₃²⁻?

In the carbonate ion CO₃²⁻, the carbon atom and the three oxygen atoms are non-metals. Non-metal atoms join by sharing electrons, so the bonds between carbon and oxygen are covalent.

The 2− charge makes the whole ion negative. It is held to metal ions such as Na⁺ or Ca²⁺ by ionic (electrovalent) bonds, but that is a bond between the ion and the metal, not inside the ion.

Metallic bonding exists only between metal atoms, so it does not apply here.