Crazy for Cats

The majority of these cats can be obtained via trading, and 7 can be purchased from vendors for 40s - 50g.
You only need to obtain 20 of the 26 listed.
 

Can Trade:

Pet

Source

Bombay Cat Also available on the BMAH
Cornish Rex Cat
Orange Tabby Cat
Silver Tabby Cat

Vendor
Alliance: Donni Anthania, Elwynn Forest 44.2, 53.2

White Kitten

Vendor
Alliance: Lil Timmy, Stormwind City

Siamese Cat Also available on the BMAH

Vendor
Neutral: Dealer Rashaad, Netherstorm 43.4, 35.2

Calico Cat

Vendor
Neutral: Breanni, Dalaran 58.6, 39.6

Winterspring Cub

Vendor
Neutral: Michelle De Rum, Winterspring 59.8, 51.6

Darkmoon Cub

World Event
Darkmoon Faire - Lhara, Darkmoon Island 48.6, 69.8

Feline Familiar

World Event
Hallow's End - Alliance: Pippi, Elwynn Forest | Horde: Woim, Undercity

Sapphire Cub

Profession
Jewelcrafting

Nightsaber Cub
Spectral Tiger Cub

TCG

Black Tabby Cat Also available on the BMAH

World Drop Zone: Hillsbrad Foothills

Guardian Cub

Blizzard Pet Store
June 2014: No longer available for purchase. Source

 
 

Cannot Trade:

Pet

Source

Xu-Fu, Cub of Xuen

Vendor
Tournament of Celestials: Master Li, Timeless Isle 34.7, 59.7

Cat
Cheetah Cub
Fluxfire Feline
Sand Kitten
Snow Cub

Pet Battles
Arathi Highlands, Elwynn Forest, Eversong Woods, Netherstorm, Silvermoon City
Northern Barrens
Gnomeregan, New Tinkertown, Dun Morogh
Tanaris
Dun Morogh

Mr. Bigglesworth

Achievement
Raiding With Leashes*
*All pets for the achievement criteria can be traded, however Mr Bigglesworth cannot be traded.

Panther Cub

Quest
Some Good Will Come (Alliance | Horde)

Cinder Kitten

Blizzard Pet Store

Brightpaw

Blizzard Pet Store

Spectral Cub

Promotion
Battle.net World Championship Shanghai 2012

Pet

Item

Source

Pet

Item

Source

[npc=7383]

[item=8491]

Drop

[npc=7385]

[item=8485]

Vendor Alliance: [npc=6367]

[npc=85283]

[item=128424]

Blizzard Store

[npc=34364]

[item=46398]

Vendor: [npc=28951]

[npc=62019]

Pet Battle

[npc=62129]

-

Pet Battle

[npc=68267]

[item=92707]

Bizzard Store

[npc=7384]

[item=8486]

Vendor Alliance: [npc=6367]

[npc=56031]

[item=74981]

Vendor: [npc=14846] for 90 [currency=515]

[npc=53884]

[item=70908]

Vendor Hallow's End

[npc=68838]

-

Pet Battle

[npc=53283]

[item=72068]

Blizzard Store

[npc=68655]

[item=93031]

Achievement Reward: Raiding with Leashes

[npc=52344]

[item=68841]

TCG

[npc=7382]

[item=8487]

Vendor Alliance: [npc=6367]

[npc=52226]

[item=68833]

Quest Reward: Some Good Will Come

[npc=62257]

Pet Battle

[npc=61883]

[item=82775]

Jewelcraft: Sapphire Cub

[npc=7380]

[item=8490]

Vendor: [npc=20980]

[npc=7381]

[item=8488]

Vendor Alliance: [npc=6367]

[npc=61689]

Pet Battle

[npc=68502]

[item=90953]

Promotion

[npc=36511]

[item=49343]

TCG and Trading Post for 750 [currency=2032] in July 2023 (not sure yet if or when it returns)

[npc=7386]

[item=8489]

Vendor Alliance: [npc=8666]

[npc=52831]

[item=69239]

Vendor: [npc=52830]

[npc=71942]

[item=101771]

Vendor: Master Li for 3 [item=101529]

You do not need all 26 pets, the achievement only requires you to have any 20 of them :-)


Update 2017-08-13: Added [npc=85283]
Update 2023-07-03: Added Trading Post source for [item=49343]

There should be a title: "the Crazy Cat Lady"

Edit: And now there is a title! :D