There are several badges a character can obtain in order to increase the amount of Inventions holding capacity they have. The only exception is no badge increases the storage capacity of Vault Reserve; that number is static, tied to character level.
Increase Salvage Storage
Each of the following Invention Salvage related badges increases the amount of on-hand Invention Salvage holding capacity by two, for a possible total of +20 on-hand Invention Salvage slots. Except for Scrounger, which is the sale of salvage on the Consignment House, the badges are for memorizing the level 25-30 IO Enhancement recipes.
Additionally, characters level 35 and above can craft the Vanguard Storage Sack temporary power for +10 Invention Salvage holding capability.
Increase Recipe Storage
Each of the following Invention Recipe related badges increases the amount of on-hand Invention Recipe holding capacity by one, for a possible total of +9 on-hand Invention Recipe slots. The badges are for memorizing the level 45-50 IO Enhancement recipes.
Each of the following Consignment House transaction slot related badges increase the number of transaction slots by one, for a possible total of +5 slots. The badges are awarded by selling any items (salvage, recipes, inspirations, etc.) on the market.