As of today, it is possible to have obtained a total of 191 Orbs without purchasing any with real money, if you logged in every day. By the time the last currently running event ends, it is predicted that this will rise to 243 Orbs. The following is a breakdown of these numbers.
You get 15 Orbs at the start of the game.
There are three missions in the "Prologue" chapter that give one Orb each for clearing them for the first time. These only appear on Normal difficulty.
|10||Link Nintendo Account
The My Nintendo rewards shop for Fire Emblem Heroes contains a free one-time gift of 10 summoning Orbs. To claim them, create a My Nintendo account (if you don't already have one), link it to the Fire Emblem Heroes app (either when you open the app for the first time or by going to Misc. -> Account Management), then go to the My Nintendo Rewards Shop and redeem the "10 Orbs" prize. The Orbs can be retrieved from the owl on the home tab of the game.
|135||Clear Story Mode (all difficulties)
Each story mode chapter gives one Orb the first time you clear it. With 9 chapters split into 5 parts and 3 difficulties, that's a total of 135 Orbs for clearing all of them. This will require a total of 1470 Stamina (267 for Normal, 489 for Hard, 714 for Lunatic). Keep in mind that the Stamina required for each mission steadily increases throughout the difficulties, with the later Lunatic missions costing 15+ Stamina each, so don't expect to clear more than a few in one sitting without burning through Stamina potions.
|2 per day||Daily Login Bonus
As part of a launch promotion, you get two Orbs the first time you log in every day, claimable from the owl. People predict this will end at the same time the current Summoning Focuses end (2/14 at 8:59 p.m.), due to the Orb event being included in their announcement. This is, however, unconfirmed, and the login bonus may end later (or even earlier) than expected.
|1 per Sunday(?)||Weekend Login Bonus
This past Sunday's special weekend login bonus was 1 Orb. It is unclear whether this will be the case for every Sunday.
|6 per week||Special Maps
As part of the launch celebration, two maps will be added to Special Maps per week, with the last planned one being aded on 3/5 at 9:00 p.m. Each map awards three Orbs upon clearing it. All maps will disappear on 3/13 at 8:59 p.m.
|9 per month||Orb Quests
Each month has special quests that can be completed for various rewards. The quests are split into three difficulties: Normal, Hard, and Lunatic. All Normal quests must be cleared to unlock the Hard quests, and all Hard quests must be cleared to unlock the Lunatic quests. Each difficulty has one quest that awards Orbs, with Normal giving one Orb, Hard giving three Orbs, and Lunatic giving five. Because of the limited Stamina and Duelling Swords, it is recommended to start working on these quests as soon as possible.