See if the following modification works for you.....

The below code will prompt you to input a date in Input Box and then check the column A in Sheet1 of all the destination workbooks and if found, it copies the row to the Master workbook.

Sub Import_to_Master()
    Dim sFolder As String
    Dim sFile As String
    Dim wbD As Workbook, wbS As Workbook
    Dim wsD As Worksheet
    Dim myDate As Variant, r As Long
         
    Application.ScreenUpdating = False
    Set wbS = ThisWorkbook
    sFolder = wbS.Path & "\"
     
    sFile = Dir(sFolder)
    Do Until IsDate(myDate) = True
    myDate = InputBox("Enter A Date (DD/MM/YYYY).")
    If Not IsDate(myDate) Then
        MsgBox "You have entered an invalid Date. Please try again.....", vbExclamation, "Invalid Date"
    End If
    Loop
    myDate = CDbl(CDate(myDate))
    Do While sFile <> ""
        If sFile <> wbS.Name Then
            Set wbD = Workbooks.Open(sFolder & sFile) 'open the file; add condition to
            Set wsD = wbD.Sheets("Sheet1")
            If WorksheetFunction.CountIf(wsD.Columns(1), myDate) > 0 Then
                 ' >>>>>> Adapt this part
                 r = Application.Match(myDate, wsD.Range("A:A"), 0)
                wsD.Range("A" & r & ":H" & r).Copy
                wbS.Sheets("Sheet1").Range("A" & Rows.Count).End(xlUp).Offset(1, 0).PasteSpecial xlPasteValues
                Application.CutCopyMode = False
                 ' >>>>>>
             End If
            wbD.Close savechanges:=True 'close without saving
        End If
        sFile = Dir 'next file
    Loop
    Application.ScreenUpdating = True
End Sub