You can buy PIAAs in a kit or single lamps. They don't all come with wiring. Yes you absolutely need to run the lamps off a rely or your switch will need to handle a lot of amps. I know those red ones you have are only rated for 6 or 10 amps max initial load.
The PIAA 510 energize at about 12 amps and run at around 7 or 8 amps. Way to much for the little switches you have there.
Think of the "load" wire as the wire that sends the signal to the relay to trip it and send the larger current to the lights. So you connect the hot lead of the switch to the source you want to draw your low amp signal from (options 1-3 below) and the "load" wire goes out to the relays signal lead.
Depending on how you want to run the "signal" to the relay the power for the switch should come from one of the following;
1. The ignition on hot source (in the internal FJ fuse box it is the empty fuse socket on the bottom row far right) if you only what the lights to be able to come on only when the key is on.
2. The always hot source (the empty fuse socket second row, second socket from the left) if you what to be able to turn on the lights with the key off or on.
3. Or like most directions say you can tap the high beam signal wire if you want the lights to come on with the switch on and you flip the high beams on. I have never done it this way in any of my vehicles so I haven't traced down the FJs high beam signal lead.
I can't find the wiring diagram I drew up for mine but I'll keep looking.