You can find here the chance for each ingredient to appear in the inventory of the alchemists of each hold.

Guaranteed: each ingredient has 100% chance to appear, numbers between 3 and 9.
Common: each ingredient has about 50% chance to appear, numbers between 2 and 9.
Uncommon: each ingredient has about 15% chance to appear, 1 sometimes 2.
Rare: each ingredient has about 5% chance to appear.
None: 0% chance.

Skyrim Alchemists Ingredients (Skyrim + Dawnguard + Hearthfire):

    Antlers (Large Antlers, Small Antlers):
        The Pale: Uncommon
        WinterHold: Uncommon
        Whiterun: Guaranteed
        Everywhere Else: Common
        
    Bear Claws:
        Everywhere: Uncommon
        
    Beehive Husk:
        The Rift: Guaranteed
        Eastmarch: Uncommon
        Falkreath: Uncommon
        Whiterun: Uncommon
        Everywhere Else: None
        
    Honeycomb:
        The Rift: Guaranteed
        Eastmarch: Uncommon
        Falkreath: Uncommon
        Whiterun: Uncommon
        Everywhere Else: None
    
    Rock Warbler Egg:
        The Reach: Guaranteed
        Everywhere Else: Uncommon
        
    Pine Thrush Egg:
        The Rift: Guaranteed
        Falkreath: Guaranteed
        Everywhere Else: Uncommon
            
    Chicken's Egg:
        Everywhere: Common
    
    Bone Meal:
        Everywhere: Common
        
    Briar Heart:
        The Reach: Uncommon
        Everywhere Else: Rare
    
    Hawk's Egg:
        Haafingar: Guaranteed
        Everywhere Else: Rare
    
    Salmon Roe:
        Everywhere: Rare    
    
    Canis Root:
        Haafingar: Guaranteed
        Hjaalmarch: Guaranteed
        The Rift: Guaranteed
        Everywhere Else: None
    
    Charred Skeever Hide:
        Everywhere: Common
        
    Chaurus Egg:
        Everywhere: Uncommon
        
    Creep Cluster:
        Eastmarch: Guaranteed
        Everywhere Else: None
        
    Bee:
        The Rift: Guaranteed
        Eastmarch: Uncommon
        Falkreath: Uncommon
        Whiterun: Uncommon
        Everywhere Else: None
    
    Small Fishes (Abecean Longfin, Cyrodilic Spadetail, Histcarp, Silverside Perch, River Betty):
        The Rift: Common
        Everywhere else: Uncommon
    
    Daedra Heart:
        Everywhere: Rare
        
    Deathbell:
        Hjaalmarch: Guaranteed
        Haafingar: Common
        Everywhere Else: None
        
    Chaurus Hunter Antennae:
        Everywhere: Uncommon
            
    Gleamblossom:
        Everywhere: None

    Ancestor Moth Wing:
        Everywhere: None
        
    Poison Bloom:
        Everywhere: None
        
    Yellow Mountain Flowers:
        Everywhere: None
        
    Dartwings (Blue Dartwing, Orange Dartwing):
        Hjaalmarch: Common
        The Pale: None
        Winterhold: None
        Everywhere Else: Uncommon
        
    Dragon's Tongue:
        Eastmarch: Guaranteed
        Everywhere Else: None
        
    Dwarven Oil:
        The Reach: Guaranteed
        Everywhere Else: Uncommon
        
    Ectoplasm:
        Everywhere: Uncommon
        
    Elves Ear:
        Everywhere: Common
        
    Falmer Ear:
        Everywhere: Uncommon
    
    Torchbug Thorax:
        Hjaalmarch: Common
        Falkreath: Common
        Whiterun: Common
        The Pale: None
        Winterhold: None
        Everywhere Else: Uncommon
    
    Fire Salts:
        Everywhere: Rare
        
    Frost Mirriam:
        Everywhere: Uncommon
        
    Frost Salts:
        Everywhere: Rare
        
    Garlic:
        Everywhere: Common
        
    Giant Lichen:
        Hjaalmarch: Guaranteed
        Everywhere Else: None
        
    Giant's Toe:
        Everywhere: Uncommon

    Glow Dust:
        Everywhere: Uncommon
    
    Glowing Mushroom:
        Everywhere: Common

    Hagravens (Hagraven Claw, Hagraven Feathers):
        The Reach: Uncommon
        Everywhere: Rare
        
    Hanging Moss:
        The Reach: Guaranteed
        Everywhere Else: Uncommon
        
    Hawk (Hawk beak, Hawk Feathers):
        Haafingar: Guaranteed
        Everywhere Else: Uncommon
        
    Human (Human Flesh, Human Heart):
        Feran Sadri: Guaranteed
        Everywhere Else: Uncommon
    
    Ice Wraith Teeth:
        The Pale: Common
        Winterhold: Common
        Everywhere Else: None
    
    Jazbay Grapes:
        Eastmarch: Guaranteed
        Everywhere Else: None
        
    Juniper Berries:
        The Reach: Guaranteed
        Everywhere Else: None
        
    Lavenders:
        Whiterun Guaranteed
        The Reach: Common
        Haafingar: Common
        Hjaalmarch: Common
        Everywhere Else: None
    
    Moonsugar:
        Khajiit Merchants: Guaranteed
        Everywhere Else: Rare
                    
    Mora Tapinella:
        The Pale: None
        Winterhold: None
        Everywhere Else: Common    

    Butterflies (Blue Butterfly Wing, Butterfly Wing, Luna Moth Wing):
        Hjaalmarch: Common
        Falkreath: Common
        Whiterun: Common
        The Pale: None
        Winterhold: None
        Everywhere Else: Uncommon            
        
    Mountain Flowers (Blue and Purple):
        The Rift: Guaranteed
        Falkreath: Guaranteed
        Eastmarch: Guaranteed
        Whiterun: Guaranteed
        Everywhere Else: Common
        
    Red Mountain Flowers:
        The Rift: Guaranteed
        Falkreath: Guaranteed
        Eastmarch: Guaranteed
        Everywhere Else: Uncommon
    
    Mudcrab Chitin:
        Everywhere: Common

    Cave Mushrooms (Bleeding Crown, White Cap, Fly Amanita, Blisterwort, Namira's Rot, Imp Stool):
        Everywhere: Common
        
    Nightshade:
        Falkreath: Guaranteed
        Winterhold: None
        Everywhere Else: Uncommon
        
    Nirnroot:
        Everywhere: Rare
        
    Crimson Nirnroot:
        Everywhere: None
    
    Nordic Barnacle:
        The Reach: None
        Whiterun: None
        Falkreath: Uncommon
        The Pale: Guaranteed
        Haafingar: Guaranteed
        Everywhere Else: Common

    Powdered Mammoth Tusk:
        Everywhere: Rare

    Pearls:
        Everywhere: Rare
    
    Sabre Cat (Sabre Cat Tooth, Eye of Sabre Cat)
        Everywhere: Uncommon
    
    Salt Pile:
        Everywhere: Common
        
    Scaly Pholiota:
        The Rift: Guaranteed
        Everywhere Else: None
    
    Skeever Tail:
        Everywhere: Common
    
    Slaughterfish (Slaughterfish Egg, Slaughterfish Scales):
        The Reach: Uncommon
        Whiterun: Uncommon
        Falkreath: Uncommon
        Everywhere Else: Common
        
    Snowberries
        The Pale: Guaranteed
        Winterhold: Guaranteed
        Eastmarch: Guaranteed
        Whiterun: Uncommon
        The Reach: Uncommon
        Everywhere Else: Common
        
    Spider Egg:
        Everywhere: Uncommon
    
    Grass Pod:
        Haafingar: Guaranteed
        Hjaalmarch: Guaranteed
        The Pale: Guaranteed
        WinterHold: Guaranteed
        Eastmarch: Common
        Everywhere Else: None
    
    Spriggan Sap:
        Everywhere: Rare
    
    Swan Fungal Pod:
        Hjaalmarch: Guaranteed
        Everywhere Else: None

    Taproot:
        Everywhere: Rare
        
    Thistle:
        Haafingar: Common
        Whiterun: Common
        Eastmarch: Guaranteed
        Falkreath: Guaranteed
        Everywhere Else: None
        
    Troll Fat:
        Everywhere: Uncommon
        
    Tundra Cotton:
        Whiterun: Guaranteed
        Everywhere Else: None
        
    Vampire Dust:
        Everywhere: Uncommon
    
    Void Salts:
        Everywhere: Rare
                
    Wheat:
        Whiterun: Common
        Eastmarch: Common
        Haafingar: Common
        The Rift: Common
        Everywhere Else: None
        
    Wisp Wrapping:
        Everywhere: Rare
        
        
Solstheim Alchemists Ingredients (Dragonborn):

    Ash Creep Cluster: Common
    Ash Hopper Jelly: Common
    Boar Tusk: Common
    Burnt Spriggan Wood: Uncommon
    Spawn Ash: Common
    Emperor Parasol Moss: Uncommon
    Netch Jelly: Common
    Scathecraw: Common
    Ashen Grass Pod: Common
    Felsaad Tern Feathers: Common
    Trama Root: Common
    Pearl: Common
    Small Pearl: Common
    Snowberries: Common
    Bone Meal: Common
    Torchbug Thorax: Common
    Mudcrab Chiting: Common
    Large and Small Antlers: Common
    Grass Pod: Common
    Bear Claw: Uncommon


Changes with V3.0:
Powdered Mammoth Tusk from uncommon to rare.
Glow Dust from rare to uncommon.
Moon Sugar added as rare ingredient in alchemist (still guaranteed on Khajit caravans).
Pearls from none to rare (still common on solstheim).
Hawk Egg from uncommon to rare (still guaranteed in haafingar).