How do you earn Supply Drops?
You can earn supply drops without spending money by...
...receiving them as a random reward at the end of a match.
...completing Contracts, timed challenges (get X kills, get X headshots, etc.) you purchase with Armory Credits (which you can earn for free on a timer or through completing certain challenges).
...completing daily and weekly orders, challenges without a time limit that you can turn in at any time, even after the daily/weekly selection refreshes. They don't require Armory Credits to purchase, but you can only hold three at once.
...watching other players open supply drops. Curiously, you'll also be able to earn supply drops and other rewards through a new social score. By interacting with players in the HQ mode, you can inspect them and give or receive a one-time commendation, which increases your social score. But again, here's the icky bit: you can also increase your social score by watching others open their supply drops. This has its own reward system, though I haven't been in a live environment long enough to test it.