If you don’t have a database to connect to (which I understood from your question), you don’t even need to have a <connectionStrings> section in Web.config . This section is only necessary if you intend to connect to the database.
If you are using a database, then connectionString varies depending on several factors, such as authentication type, database product (MS SQL Server, MySQL), driver type (ODBC, .NET), etc.
The "vendor name" will depend on the database product you are using. For example, for SQL Server "System.Data.SqlClient"
You can look at this site for a complete list of database products and connection strings suitable for each product for different types of authentication, drivers used, etc.
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