The inductor saturation current must be greater than the charge current plus 1/2 the ripple current. You don’t want the current to get close to the saturation current of your inductor or the circuit will fail and the chip will be destroyed. Your duty cycle, D= 67.5% (8.1V/12V), so your inductor ripple current would be 1.33A.
Iripple = {Vin x D x (1-D)}/(fs x L) = {12 x 0.675 x (1-0.675)}/(600kHz x 3.3uH) = 1.33A
So the inductor should be rated for greater than 10A + 1/2(1.33A) = 10.67A. The evaluation board for the series uses Vishay IHLP5050EZERxxxM01 series, so I would recommend using the IHLP5050EZER3R3M01, which is rated for a saturation current of 32A, which gives you plenty of headroom.
Section 8.2.2.2 refers to the input capacitor. The ripple current for the input capacitor is going to be about 4.7A. They recommend using about 20uF low ESR type capacitors like ceramic, something like a pair of CGA8P2X7R1E106K250KA would be a good choice.
You do need to worry about the output capacitance and the LC filter as described in section 7.3.6. The LC filter needs to have a resonant frequency between 12kHz and 17kHz for best stability. Table 5 on page 20 recommends 40uF of capacitance along with the 3.3uH inductor, which gives a resonant frequency of about 13.85kHz. They recommend ceramic capacitors for the output as well, so you could use four of the same CGA8P2X7R1E106K250KA there. Ripple current on the output capacitors will be under 1/2A so they will handle that easily and give only a small ripple voltage.