Which Fallout game was it first stated that vaults were designed to conduct experiments?
The idea of using the Vaults as social experiments was created by Tim Cain and was touched upon in Fallout 2. While reading the Vault 8 records, you can unlock a special file with a successful science roll. It shows that the vault was a'Control Vault'. Designed to remain sealed for 10 years then begin to re-colonise the surface. This site has some good information under the 'Vault System' section.
Here is a fallout wiki link to the known vaults, what game they were encountered in and the reason for their creation.