I’ve got plenty of gold and want the RAF rewards; I was thinking of seeking out randoms and giving away 10x a month of game time using the recruit a friend system in order to get that sweet mog outfit.
However, with how much things have changed with the Battle Chest stuff I’m not even sure if I can do this, or what I would need to gift them and how. What needs to be purchased for a new RAF player so they can actually play and we get credit towards the pyramid sche-- I mean, mog stuff.
All they need is game time – that grants them access to everything up to & including Legion content.
I recruited myself to get all 12 rewards and all my accounts just had the base game (i.e., just up through Legion).
Be aware that you get rewards ONCE per month PER eligible recruit (i.e., one with game time on their account). So if you only recruit one account then it would take 12 months to get all the rewards. But if you recruit multiple then it goes faster. Also, your account must have game time on it to receive the rewards.
As far as to how to give the time, you can gift time to other people via the Shop gift option for game time.
So, if your location permits it – then you can purchase Tokens from the AH and convert them to Battle Net Balance and use that Balance to purchase the game time.
If your location does not permit the conversion to Balance, then you would just need to gift the player the gold directly* so that they could purchase their own Token for game time.
*I would caution you on ensuring that you and the other player(s) are all on the same page about what is expected.
Same, though I haven’t done all 12 rewards yet.
While the second account is active, I’ve often dual-boxed it on one machine - two instances of WoW are easily handled by even my crappy 6-year-old toaster - solely for gathering mats. The double income is quite nice, especially when I’d otherwise just let the second account sit.
I get what OP is aiming for, and that’s awesome to want to gift a random person with game time! Just offering my two cents and another option. 
Thanks for the replies thus far. I have a follow-up question now that I’ve learned I can only recruit 4 people per month. Can I instead gift 4 people 3x months right now and get all the rewards in three months from now?
Yes, that should do it.
Yes. You’ll be able to receive one reward per linked account per month, so you’ll be able to receive the first four rewards during the first month, the next four the second and the last four in month three.