Websites generally use a single login for ALL website users. You do not want to create a login for every web user in SQL Server.
The solution to your problem is to use the SESSION_CONTEXT to store the web user before calling SQL Server. Then you can retrieve it in your code.
Before every SQL call, run (yes, you need to do this before every call because the connection may be pooled):
EXEC sys.sp_set_session_context @key = N'UserName', @value = '{This is the user name}';
Then in your SQL Code you use:
SESSION_CONTEXT(N'UserName')