UPDATE: Mumper confirmed on Twitter that we will get one choice of elemental spirit from the main quest. After completing it, dailies will unlock for each spirit (air, earth, fire, water) and each Pandaren elemental spirit will have a chance to drop from its daily's bag.
Mumper also mentions that the elemental spirit quest and dailies will only be available "after you finish the last quest on the main pet battle quest line."
So better make sure you have all the initial quests completed come 5.1. ;)
Thanks to a heads up from Goldacc, I've found a possible source for the Pandaren elemental spirit pets (Pandaren Air Spirit, Pandaren Earth Spirit, Pandaren Fire Spirit, Pandaren Water Spirit).
Wowhead has them listed as a reward for a one time quest, Pandaren Spirit Tamer.
There's a catch though: you can only choose one as a reward.
But! Don't panic yet. It looks as though the quest becomes a daily after completing it, and rewards the Sack of Pet Supplies. My guess is that the sack has the chance to drop the remaining Pandaren elemental spirits that you weren't able to pick initially.
At least I very much hope so! It would be devastating if collectors could only obtain one out of the four. :(
Both quest and dailies are account-wide as well, so it's not as if we could hop onto alts, do the main quest again and choose a different elemental reward.
Keep in mind that this is still all in the testing phase and things could change. Data from Wowhead could also be off or inaccurate. We'll have to wait and see how things actually go.
Sidenote: I'm pretty bummed that I can't get onto the PTR to take a look at all of this firsthand. :( *sigh*