Do I need a database license for my application?
It does not sound like your database and application would be very big in current terms. 

You could look into the freely available Microsoft tools for Visual Studio Express and SQL Server Express:

https://www.visualstudio.com/en-us/products/visual-studio-express-vs.aspx

http://www.microsoft.com/en-us/server-cloud/products/sql-server-editions/sql-server-express.aspx

Both of these are without charge and contain most features of the bigger products.  The limit is in database size, amount of memory, and number of processors that can be used.