Forms don't use DSN; data do.  So check all data sources (tables, queries) for the form to see if they have the proper connections built-in.  Do you use the Windows ODBC app to set up these connections (see picture below)?  Check if the correct and up-to-date ODBC driver is used.  I've fixed many connection issues by just updating the driver.   Using the ODBC app is better than using a file DSN.  Storing passwords and credentials in a plain text file is probably not a good idea.

Thank you Kevin. I am Following Richards recommendations creating a File ODBC.  I am still using Access32Bit.
This is what I am doing
Private Sub RelinkTablesBtn_Click()

    'Status "Connecting to Server..."
    RelinkTable "EntityT", "EntityID"
    RelinkTable "ContactsT", "contactID"
    RelinkTable "DocumentT", "DocumentID"
    RelinkTable "HelperT", "HelperID"
    RelinkTable "HelperTypeT", "HelperTypeID"
    RelinkTable "SearchFieldT", "ID"
    RelinkTable "tblEmail_Attachments", "Email_Attachments_ID"
    RelinkTable "tblEmails", "Email_ID"
    RelinkTable "tblUser", "ID"
      
End Sub
'---------------------------------------------------------------------

Private Sub RelinkTable(TableName As String, IndexField As String)

    Dim db As Database
    Dim td As TableDef
    Dim SQL As String
    
    Status "Relinking " & TableName
    
    Set db = CurrentDb
    On Error Resume Next
    db.TableDefs.Delete (TableName)
    On Error GoTo 0
    
    Set td = db.CreateTableDef(TableName)
    td.Connect = TempVars("ConnectStringWithPW")
    td.SourceTableName = TableName
    db.TableDefs.Append td
    
    'Application.SetHiddenAttribute acTable, TableName, True
    
    SQL = "CREATE UNIQUE INDEX " & IndexField & "_index ON " & _
        TableName & " (" & IndexField & ")"
    db.Execute SQL

End Sub

This is straight out of Richards Code.  
Can anyone see something missing?

I found that the Problem was that a pause is needed in between the Relinks.
I added Sleep 1000 and that apparently fixed the Problem.
This makes me think also that In the Relink sub we are not clearing the variables.

Private Sub RelinkTablesBtn_Click()

    'Status "Connecting to Server..."
    RelinkTable "EntityT", "EntityID"
    Sleep 1000
    RelinkTable "ContactsT", "contactID"
    Sleep 1000
    RelinkTable "DocumentT", "DocumentID"
    Sleep 1000
    RelinkTable "HelperT", "HelperID"
    Sleep 1000
    RelinkTable "HelperTypeT", "HelperTypeID"
    Sleep 1000
    RelinkTable "SearchFieldT", "ID"
    Sleep 1000
    RelinkTable "tblEmail_Attachments", "Email_Attachments_ID"
    Sleep 1000
    RelinkTable "tblEmails", "Email_ID"
    Sleep 1000
    RelinkTable "tblUser", "ID"
      
End Sub

Update:  The Problem still Persist.
Richard, please take a look when you can.
Thanks

Your connection string is stored in a TempVars, would it be being cleared somehow? Perhaps try using Constants instead? Just brainstorming.

Kevin, Thank you.
I believe you are right. Somehow the "tempvars" are getting reset.
I changed the Tempvars Connection Strings to Global variables and it is working fine for now.
I will report any changes.

One more thing: Do you think that
RelinkTable "tblUser", "ID"
may have a problem with the use of the index word "ID"? (Reserved?)

whare:
Private Sub RelinkTable(TableName As String, IndexField As String)

    Dim db As Database
    Dim td As TableDef
    Dim SQL As String
    
    Status "Relinking " & TableName
    
    Set db = CurrentDb
    On Error Resume Next
    db.TableDefs.Delete (TableName)
    On Error GoTo 0
    
    Set td = db.CreateTableDef(TableName)
    td.Connect = ConnectStringWithPW
    td.SourceTableName = TableName
    db.TableDefs.Append td
    
    'Application.SetHiddenAttribute acTable, TableName, True
    
    SQL = "CREATE UNIQUE INDEX " & IndexField & "_index ON " & _
        TableName & " (" & IndexField & ")"
    db.Execute SQL

End Sub