Assuming this is still an issue here are a few things to try
sql = "SELECT * FROM [tbl_KGA_MEMBER_DETAILS] WHERE [F_Name] LIKE '" & myLetter _
& "%' ORDER BY [F_Name] ASC"
Note using string concatenation is a huge sql injection risk, but I don't remember if you can use parameterised queries with VB6 (and I haven't had it installed for years).
If you were writing the query in Access you would use * for the wildcard but using OleDb I think you need to use the sql-like '%' for the wildcard. You could always removed the WHERE altogether until you track down the problem.
If that doesn't work try putting
Kga_RS.MoveFirst
before the loop
If that still doesn't work then look again at your connection string. Try
Set DBcON = New Connection
Dim strConn As String
strConn = "Provider=Microsoft.Jet.OLEDB.4.0;Data Source=" & dbFile & ";"
conn.Open (strConn)
Finally, you have not said where App.Path is - if that is (e.g.) Program Files (x86) then any data added to the database will probably be in the users AppData folder and not in the App.Path (Assuming you are using a version of Windows >= Vista) - try moving the database to an explicit folder (like C:\temp for example)