For the Order of the Awakened, it seems as though there's only one main way to earn rep - complete the daily quest Unseen Influence for 1500 reputation.
You won't want to miss a day since you need Exalted with the faction before you can buy the Blazing Firehawk.
At the rate of 1500 per day, it'll take a total of 28 days of dailies. If you're lucky enough to come across a Medallion of the Legion it will cut down your grind time by roughly one day per medallion.
The Saberstalkers might be an easier rep grind, if you don't mind killing (elite) mobs over and over that is. You need to defeat blackfang NPCs in Fang'rila for their claws (the currency to buy Savage Cub) anyway though. Only Revered is necessary and you'll need 1500 claws.
It's tough to estimate how long it will take to reach Revered since there's a few different mobs that you can kill. The majority will reward 30 rep each, while the Fang'rila rare elites give 600 a kill. On top of that there's a weekly quest that rewards 3500 rep.
If you complete just the weekly, it'll take you 6 weeks to reach Revered. But it's almost certain that the grind time will be less since you have to kill the NPCs for claws and will earn reputation alongside that.
And of course, the Medallion of Legion mentioned above can be used for this faction too.
Keep in mind that all of these values are before any reputation bonuses, such as the buff from the level 3 Trading Post. I don't know if Order and Saberstalker reputation is affected by rep modifiers, but I'm hoping so!