Driving IRF9530 MOSFET with 5V
Can the FET can supply require current with the given gate voltage?

Not really, you do not have a lot of voltage margin left.

From the left plot I can see that at \$I_D\$ = 3 A and \$V_{gs}\$ = -5 V the FET's \$V_{DS}\$ will be about -1 V.

That leaves 5 V - 1 V = 4 V for the LEDs + series resistor. If the LEDs actually have a 3 V forward voltage then this might just work but often LEDs do have a higher forward voltage especially White LEDs.

You could try to decrease the 27 ohms resistors such that you do get the required current. That will still not eliminate the 1 V drop across the MOSFET though.

This MOSFET is really better suited to a much higher \$V_{GS}\$ like 10 V as then the drop would be about 0.5 V, that's still a lot!

Also note that if you manage to get the 3 A and the MOSFET drops 1 V it will dissipate 3 W so it will get hot! It will need a heatsink!

You really need to consider using a "better" MOSFET with a much lower \$R_{DSon}\$, for example the AO3401. Since the 3.2 A is on the edge for this device I'd use 2 of them and let each power half of the LEDs.