The photorecombination rate coefficients of potassium-like Fe VIII ions forming calcium-like Fe VII and of argon-like Fe IX forming potassium-like Fe VIII were measured by employing the merged electron-ion beams method at the Heidelberg heavy-ion storage-ring TSR. New theoretical calculations with the AUTOSTRUCTURE code were carried out for dielectronic recombination (DR) and trielectronic recombination (TR) for both ions. We compare these experimental and theoretical results and also compare with previously recommended rate coefficients. The DR and TR resonances were experimentally investigated in the electron-ion collision energy ranges 0-120 eV and 0-151 eV for Fe VIII and Fe IX. Experimentally derived Fe VIII and Fe IX DR + TR plasma rate coefficients are provided in the temperature range kBT=0.2 to 1000eV. Their uncertainties amount to ±26% and ±35% at a 90% confidence level for Fe VIII and Fe IX, respectively.