Write out the Ka and Kb for the biocarbonate ion with chemical equation:

Would this be correct:

HCO3^-(aq) <--> H+(aq) + CO3^-2(aq)

Ka = [H+][CO3-2]/[HCO3-
H2O + CO3-2(aq) <--> HCO3-(aq) + OH-(aq)

Kb = [HCO3-][OH-]/[CO3-2]

Or in my textbook there a chart that says:

Ionization Constants of Some Polyproctic Acids:

ka= H2C03 +H20<---> H30 + HCO3-
KB= HCO3-+ H20 <--->H30^+ + C03^2-

You have written so much information without identifying it that I have trouble knowing what you intended to write.

H2CO3 does have two ionization constants, both k1 and k2 but that isn't what the original question asks.
You have the Ka for HCO3^- ok
That is as you have it written, both for the ionization and for the Ka expression.
For HCO3^- as a base, it should be
HCO3^- + H^+ ==> H2CO3
or if the hydolysis equation in which HCO3^- acts as a base it is
HCO3^- + HOH ==> H2CO3 + OH^-

Both of the equations you wrote out are chemically incorrect. The correct chemical equation for the bicarbonate ion (HCO3-) is:

HCO3-(aq) + H2O(l) <--> H3O+(aq) + CO3^2-(aq)

The equilibrium constant expression for the acid dissociation reaction (Ka) is:

Ka = [H3O+][CO3^2-] / [HCO3-]

The equilibrium constant expression for the base dissociation reaction (Kb) is:

Kb = [HCO3-][H3O+] / [CO3^2-]

The chart in your textbook provides the correct ionization constants for the bicarbonate ion. The Ka and Kb values given in the chart are for the same reaction, just written in a different format. So, both the equations you wrote and the chart in your textbook provide the correct ionization constants for the bicarbonate ion.