You have to open, or create, the documents to process and process them separately e.g.

Private Sub CommandButton1_Click()

Dim oRng As Range
Dim oDoc1 As Document
Dim oDoc2 As Document
Dim oDoc3 As Document

    'create (or open) the documents
    Set oDoc1 = Documents.Add("C:\Path\Template1.dotx")
    Set oDoc2 = Documents.Add("C:\Path\Template2.dotx")
    Set oDoc3 = Documents.Add("C:\Path\Template3.dotx")

    'process each document
    'fname
    If oDoc1.Bookmarks.Exists("fname") = True Then
        Set oRng = oDoc1.Bookmarks("fname").Range
        oRng.Text = TextBox1.value
        oDoc1.Bookmarks.Add "fname", oRng
    End If
    If oDoc2.Bookmarks.Exists("fname") = True Then
        Set oRng = oDoc2.Bookmarks("fname").Range
        oRng.Text = TextBox1.value
        oDoc2.Bookmarks.Add "fname", oRng
    End If
    If oDoc3.Bookmarks.Exists("fname") = True Then
        Set oRng = oDoc3.Bookmarks("fname").Range
        oRng.Text = TextBox1.value
        oDoc3.Bookmarks.Add "fname", oRng
    End If

    'lname
    If oDoc1.Bookmarks.Exists("lname") = True Then
        Set oRng = oDoc1.Bookmarks("lname").Range
        oRng.Text = TextBox2.value
        oDoc1.Bookmarks.Add "lname", oRng
    End If
    If oDoc2.Bookmarks.Exists("lname") = True Then
        Set oRng = oDoc2.Bookmarks("lname").Range
        oRng.Text = TextBox2.value
        oDoc2.Bookmarks.Add "lname", oRng
    End If
    If oDoc3.Bookmarks.Exists("lname") = True Then
        Set oRng = oDoc3.Bookmarks("lname").Range
        oRng.Text = TextBox2.value
        oDoc3.Bookmarks.Add "lname", oRng
    End If

    'address
    If oDoc1.Bookmarks.Exists("address") = True Then
        Set oRng = oDoc1.Bookmarks("address").Range
        oRng.Text = TextBox3.value
        oDoc1.Bookmarks.Add "address", oRng
    End If
    If oDoc2.Bookmarks.Exists("address") = True Then
        Set oRng = oDoc2.Bookmarks("address").Range
        oRng.Text = TextBox3.value
        oDoc2.Bookmarks.Add "address", oRng
    End If
    If oDoc3.Bookmarks.Exists("address") = True Then
        Set oRng = oDoc3.Bookmarks("address").Range
        oRng.Text = TextBox3.value
        oDoc3.Bookmarks.Add "address", oRng
    End If

    'dob
    If oDoc1.Bookmarks.Exists("dob") = True Then
        Set oRng = oDoc1.Bookmarks("dob").Range
        oRng.Text = TextBox4.value
        oDoc1.Bookmarks.Add "dob", oRng
    End If
    If oDoc2.Bookmarks.Exists("dob") = True Then
        Set oRng = oDoc2.Bookmarks("dob").Range
        oRng.Text = TextBox4.value
        oDoc2.Bookmarks.Add "dob", oRng
    End If
    If oDoc3.Bookmarks.Exists("dob") = True Then
        Set oRng = oDoc3.Bookmarks("dob").Range
        oRng.Text = TextBox4.value
        oDoc3.Bookmarks.Add "dob", oRng
    End If

    'amount
    If oDoc1.Bookmarks.Exists("amount") = True Then
        Set oRng = oDoc1.Bookmarks("amount").Range
        oRng.Text = TextBox5.value
        oDoc1.Bookmarks.Add "amount", oRng
    End If
    If oDoc2.Bookmarks.Exists("amount") = True Then
        Set oRng = oDoc2.Bookmarks("amount").Range
        oRng.Text = TextBox5.value
        oDoc2.Bookmarks.Add "amount", oRng
    End If
    If oDoc3.Bookmarks.Exists("amount") = True Then
        Set oRng = oDoc3.Bookmarks("amount").Range
        oRng.Text = TextBox5.value
        oDoc3.Bookmarks.Add "amount", oRng
    End If

    'period
    If oDoc1.Bookmarks.Exists("period") = True Then
        Set oRng = oDoc1.Bookmarks("period").Range
        oRng.Text = TextBox6.value
        oDoc1.Bookmarks.Add "period", oRng
    End If
    If oDoc2.Bookmarks.Exists("period") = True Then
        Set oRng = oDoc2.Bookmarks("period").Range
        oRng.Text = TextBox6.value
        oDoc2.Bookmarks.Add "period", oRng
    End If
    If oDoc3.Bookmarks.Exists("period") = True Then
        Set oRng = oDoc3.Bookmarks("period").Range
        oRng.Text = TextBox6.value
        oDoc3.Bookmarks.Add "period", oRng
    End If

    Unload Me
End Sub