You won't be able to get an actual extra backpack until very late in the game, but you can get almost any tailor to make you extra bags that you can equip in your bag slots.
Key:
(x) - number of slots
b - binds on pickup
u - unique
+ - available only in an instance
Crafted
Linen Bag (6)
Red Linen Bag (6)
Kodo Hide Bag (6)
Woolen Bag (8)
Green Woolen Bag (8)
Red Woolen Bag (8)
Small Silk Pack (10)
Green Silk Pack (10)
Black Silk Pack (10)
Mageweave Bag (12)
Red Mageweave Bag (12)
Runecloth Bag (14)
Mooncloth Bag (16)
Bottomless Bag (18)
Vendor-Sold Bags
Small Brown Pouch (6)
Brown Leather Satchel (8)
Heavy Brown Bag (10)
Huge Brown Sack (12)
Looted Standard Bags
Small Red Pouch (6)
Small Blue Pouch (6)
Small Green Pouch (6)
Small Black Pouch (6)
Jewelry Box (6b)
Old Moneybag (6b)
Handmade Leather Bag (6b)
Red Leather Bag (8)
Blue Leather Bag (8)
White Leather Bag (8)
Green Leather Bag (8)
Large Moneybag (8b)
Gnoll Hide Sack (8u)
Large Blue Sack (10)
Large Brown Sack (10)
Large Green Sack (10)
Large Red Sack (10)
Large Rucksack (10)
Snakeskin Bag (10u+)
Fel Steed Saddlebags (10u+)
Murloc Skin Bag (10u+)
Large Knapsack (12)
Sturdy Lunchbox (12)
Raptor Skin Pouch (12u)
Librarian's Satchel (12b)
Agmond's Belt Pouch (12ub)
Troll-hide Bag (14)
Journeyman's Backpack (14)
Traveler's Backpack (16)
Demon Hide Sack (16+)
Wayfarer's Knapsack (16+)
Quest Rewards
Handmade Leather Bag (4u)
Old Blanchy's Feed Pouch (4u)
Wizbang's Gunnysack (8u)
Captain Sander's Booty Bag (8u)
Ooze-covered Bag (10u)
Deviate Hide Pack (10u)
Gunnysack of the Night Watch (10u)
Explorer's Knapsack (14u)
Thawpelt Sack (14u)