Hi,
I changed your sub a little bit, should do what you want!

Sub Enter_Agreement_details()
    Dim Iput As String, Data As Variant, i As Long, iUB As Long, Answers() As String
Data = Array("Agreement Number", "Parent Company", "Title", "First Name", _"Surname", "Street", "Town", "City", "County", "Post Code", "Phone", "Fax", 
     "E-Mail", "Application Received", "UNA To Site", "UNA Received From Site", _
     "UNA To DEFRA", "DEFRA Approved")
iUB = UBound(Data)
    ReDim Answers(iUB)
For i = 0 To iUB
        Iput = InputBox(".:: p l e a s e  e n t e r  f o l l o w i n g  d a t a ::.  " _
         & Chr(10) & Chr(10) & Data(i), Data(i))
        Answers(i) = Iput
    Next i
With Sheet3.Range("A1").End(xlDown)
     Range(.Offset(1, 0), .Offset(1, iUB)) = Answers
    End With
End Sub
Matt