Is there a way to increase your inventory slots?

Solution 1:

There is no way to increase your inventory size beyond the 8 slots.

From http://en.wikipedia.org/wiki/Realm_of_the_Mad_God:

A character's inventory capacity is very limited as the player can store 8 items on a character (beyond what the character has equipped), plus a free private vault chest in the nexus which stores items across deaths/characters.

Players start with 1 free vault and can purchase more vault chests. Items in these chests are account-based so any items in the vault can be used to equip a new character.

http://realmofthemadgod.wikia.com/wiki/The_Nexus says:

Each chest can hold eight items; any items left on the floor will disappear when you leave your Vault. You get one vault to start off with, but the rest you must pay for using the money in the game.

Solution 2:

Well, you could buy a backpack. It is consumed on use, and doubles inventory space.