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
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 & ";"
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)