As fabian's answer points out, darts are not ammunition and do not have a chance of being lost, unless you toss one into lava or something. Unfortunately, AL only allows you to buy ammunition in cheaper bundles, so darts need to be purchased normally. That aside, there are several ways to optimize for minimal darts needed.
Eldritch Knight: Weapon Bond
At level 3, an Eldritch Knight fighter gains the Weapon Bond feature. Once you spend an hour bonding to a weapon, you are able to recall it to your hand (PHB, p. 75):
If [the bonded weapon] is on the same plane of existence, you can summon that weapon as a bonus action on your turn, causing it to teleport instantly to your hand.
You can have up to two bonded weapons, but can summon only one at a time with your bonus action.
With Eldritch Knight 3 / Rogue X, you can throw a dart with your action, and then recall it with your bonus action and not lose out on any attacks. In situations where you don't need magical damage, you can instead use your bonus action to Hide. Eldritch Knight 3 also lets you pick up the Archery fighting style (+2 to hit with your darts), shield proficiency, Action Surge, and useful spells, like find familiar and shield.
Way of the Kensei: One with the Blade
At level 6, a Way of the Kensei monk gains the One with the Blade feature (XGtE, p. 34-35):
Magic Kensei Weapons. Your attacks with your kensei weapons count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.
If you choose darts as one of your kensei weapons, all of your darts are magical. You can still benefit from the +X to hit and damage from magic items, though, so you'll want to pick some up anyway. Way of the Kensei also grants the Kensei's Shot feature at level 3 (XGtE, p. 34):
Kensei's Shot. You can use a bonus action on your turn to make your ranged attacks with a kensei weapon more deadly. When you do so, any target you hit with a ranged attack using a kensei weapon takes an extra 1d4 damage of the weapon's type. You retain this benefit until the end of the current turn.
Combine Kensei's Shot with monk's scaling Martial Arts damage, and your basic darts will one day deal 1d10+1d4+5. Since you need to take monk to level 6, you're probably better off staying pure monk if you go the Kensei route.
An aside on throwing multiple darts per turn
Monk grants you Extra Attack, allowing you to throw two darts with your Attack action. Drawing a weapon counts as your one free object interaction for the round, so here's how most combats will go:
- Start with a dart in each hand.
- Turn 1. Throw two darts. Use your object interaction to draw one dart.
- Turn 2. Draw one dart. Throw two darts.
- Turn 3+. Draw one dart. Throw one dart.
By turn 3, you're down to only one dart attack. The third bullet of the Dual Wielder feat is a (costly) way to solve this problem (PHB, p. 165):
• You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one.
On the other hand, most combats are winding down by turn 3, so maybe that's your cue to get in close and punch someone. If you're holding a dart when you do so, Way of the Kensei's Agile Parry grants you +2 to AC.
Forge Domain: Blessing of the Forge
At level 1, a Forge Domain cleric gains the Blessing of the Forge feature (XGtE, p. 19; emphasis mine):
[...] At the end of a long rest, you can touch one nonmagical object that is a suit of armor or a simple or martial weapon. Until the end of your next long rest or until you die, the object becomes a magic item, granting [...] a +1 bonus to attack and damage rolls if it's a weapon.
Once you use this feature, you can't use it again until you finish a long rest.
This increases your dart stockpile by one. By character level 5, you'd have two magical darts (one purchased, one created); by halfway through level 7, you'd have three. Hopefully whatever you're throwing darts at is dead by the third round of combat, so you can retrieve your darts from their corpse mid-battle.
Cleric 1 also lets you pick up shield proficiency, as well as useful spells, like bless, healing word, and protection from evil and good. Due to the limited dart count (and no way to recall them), Forge Cleric 1 / Rogue X is probably the optimal path here.
Horizon Walker: Planar Warrior
At level 3, a Horizon Walker ranger gains the Planar Warrior feature (XGtE, p. 42):
As a bonus action, choose one creature you can see within 30 feet of you. The next time you hit that creature on this turn with a weapon attack, all damage dealt by the attack becomes force damage, and the creature takes an extra 1d8 force damage from the attack.
Your pure force damage darts will be (almost) never resisted. Since you can only convert one attack to force damage, Horizon Walker 3 / Rogue X is the way to go. Horizon Walker 3 also lets you pick up the Archery fighting style, shield proficiency, and useful spells, like absorb elements, goodberry, and protection from evil/good.
Damage isn't everything, but comparing it will give a decent idea of a build's combat capabilities. For this comparison, I will use level 8 characters attacking an AC 16 enemy with resistance to bludgeoning, piercing, and slashing from nonmagical weapons. Damage will be average DPR over 3 rounds. Each character has one +1 dart and one +2 dart, in addition to any provided by their build. All rogue levels are using the base class features only.
- Rogue 8. The baseline; out of magic darts in two rounds. 14.1 DPR.
- Eldritch Knight 3 / Rogue 5. One +2 dart attack per round, forever. 15.8 DPR.
- Way of the Kensei 8. A +2 and +1 dart on round 1; two magical +0 darts on round 2; one magical +0 on round 3. 12.6 DPR. If you're willing to make unarmed strikes on round 3, this goes up to 15.4 DPR.
- Forge Domain 1 / Rogue 7. One +2 dart on round 1; +1 darts on rounds 2 and 3. 15.4 DPR.
- Horizon Walker 3 / Rogue 5. One force damage +2 dart on round 1; one force damage +1 dart on round 2; one force damage normal dart on round 3. 17.8 DPR.
(I used this AnyDice program to run the numbers, if you're interested. The Summary tab is the best way to view the results.)
Horizon Walker seems like the clear winner, but it isn't actually that simple. Planar Warrior requires your bonus action, which the pure rogue and the Forge Domain multiclass can use to Hide. There isn't always a place to Hide, so its usefulness will be adventure-dependent. The Eldritch Knight can have a familiar use the Help action for (generally) more reliable advantage, raising it to 19.5 DPR.
Lastly, the Sharpshooter feat will modify damage differently for each build, in exchange for less reliability. Pure rogue, Forge Domain, and Horizon Walker deal less damage with Sharpshooter, unless they have advantage. Eldritch Knight gets a small boost to 16.6 DPR, even without advantage, thanks to always using a +2 dart. Way of the Kensei gets a significant boost to 18.3 DPR, as long as you're alright with punching things on round 3.
Eldritch Knight 3 / Rogue X is my pick for most economical magical dart thrower. It only requires one magical dart, which you can throw every turn, no matter how many rounds of combat you face. You can choose to use Sharpshooter (optimally as a Variant human) when you have advantage for a large damage boost.
EK also grants you excellent defenses; shield proficiency and the shield spell combine to spike your AC to 23+ when you need it most. Find familiar grants you advantage and risk-free scouting. Absorb elements stacks with rogue's Evasion for near-immunity to Dex-based damage spells. Oh, and summoning darts to your hand is super cool.