You need to have web.config files inside the admin and vendors folder separately. Inside these web.config files you would declare your <formsauthentication> elements independently. The global web.config should contain no <formsauthentication> element.
Edit:
I will provide a xml snippet below, but implementing FormsAuthentication is not a task I would recommend for someone who is a beginner. First you should read and understand the process involved in implementing at least a basic FormsAuthentication model. That being said, this would be the relevant web.config entry under the <system.web> section:
<authentication mode="Forms">
<forms loginUrl="~/vendors/login.aspx"
protection="All"
timeout="30"
name=".ASPXAUTH"
requireSSL="false"
slidingExpiration="false"
defaultUrl="~/vendors/default.aspx"
cookieless="UseDeviceProfile"/>
</authentication>