In ICP-OES you get emission lines from both atoms and ions, depending on how easy it is to ionize atoms in each element. But the argon plasma is in the neighborhood of 104 K. A flame temperature is much lower (for Bunsen burner, about 1770 K), and maybe not enough to ionize even easily ionizable elements. Quick math supports this.
The first ionization energy of sodium is 5.139 eV, or about 8.2 x 10-19 J. For a Bunsen flame at 1770 K, kT ~ 2.4 x 10-20 J. For an ICP plasma at 10,000 K, kT ~ 1.4 x 10-19 J. The ICP plasma is much more likely to result in sodium ion emission lines because kT is much closer to the ionization energy. In fact an ionization buffer (usually cesium) is added during ICP-OES experiments to reduce ionization, which can skew measurements.
Potassium is easier to ionize than sodium, and cesium easier to ionize than potassium. So even in a Bunsen flame you may start to see some emission from ions as you go down columns in the periodic table. Still, the flame temperature is pretty low, so even for cesium (ionization energy ~ 3.9 eV, 6.2 x 10-19 J) atomic emission will likely dominate.