So I fought and fought with this, ultimately going back to one of the sample scripts I was given. Here's where I ended up with (two read variables, one write variable):

Public Sub Main()

Dim FilePathName As String = Dts.Variables.Item("InboundPathAndName").Value

Dim url As System.Uri

Dim webReq As HttpWebRequest

Try

url = New System.Uri(Dts.Variables.Item("HttpPostURL").Value.ToString)

webReq = CType(WebRequest.Create(url), HttpWebRequest)

'POST Data

webReq.Method = "POST"

webReq.ContentType = "multipart/form-data; boundary=xyz"

webReq.Headers.Add("Cookie", "SMCHALLENGE=YES")

' convert username:password to basic 64 format and append to the HTTP

' header

webReq.Headers.Add("Authorization", _

Dts.Variables.Item("HttpAuth").Value.ToString)

webReq.KeepAlive = True

Dim dataBoundary As String = "------xyz"

Dim endingBoundary As Byte() = _

System.Text.Encoding.Default.GetBytes(vbCrLf + "------xyz--" + vbCrLf)

Dim ReadIn As FileStream

Dim tempStream As Stream

'create the information we need to send as part of post to let

'the ASPX page know about the file data

Dim DataString As StringBuilder = New StringBuilder

DataString.Append(dataBoundary + vbCrLf)

DataString.Append("Content-Disposition: form-data; name=" + _

"""" + "file" + """" + "; filename=" + """" + _

FilePathName.ToString + """" + vbCrLf)

'set the file type to octet-stream so we can handle any kind of Data()

DataString.Append("Content-Type: application/octet-stream" + _

vbCrLf + vbCrLf)

'open the file to post

ReadIn = New FileStream(FilePathName, FileMode.Open, FileAccess.Read)

ReadIn.Seek(0, SeekOrigin.Begin) 'move to the start of the file

Dim FileData(1024) As Byte 'read the file in 1k chunks

Dim DataRead As Integer = 0

tempStream = webReq.GetRequestStream()

'send the data about the file

Dim FileInfo As Byte() = _

System.Text.Encoding.Default.GetBytes(DataString.ToString())

tempStream.Write(FileInfo, 0, FileInfo.Length)

Do

DataRead = ReadIn.Read(FileData, 0, 1024)

If (DataRead > 0) Then 'we have data

tempStream.Write(FileData, 0, DataRead)

Array.Clear(FileData, 0, 1024) 'clear the array

End If

Loop While (DataRead > 0)

' send the closing boundry

tempStream.Write(endingBoundary, 0, endingBoundary.Length)

'close the stream

ReadIn.Close()

tempStream.Close()

'Get the response from the server

Dim webResp As HttpWebResponse = webReq.GetResponse()

Dim sr As New StreamReader(webResp.GetResponseStream())

'put the stream data in a string

Dim respData As String = sr.ReadToEnd()

sr.Close()

webResp.Close()

'MessageBox.Show(respData) 'Troubleshoot response

Dts.Variables.Item("WebsiteResponse").Value = respData.ToString

Catch webExcp As WebException

'If you reach this point, an exception has been caught.

'Write out the WebException message.

MessageBox.Show("An error occured." + webExcp.ToString())

Return

Catch myExcp As Exception

MessageBox.Show("A WebException has been caught." + myExcp.ToString())

Return

End Try

Dts.TaskResult = ScriptResults.Success

End Sub

End Class

Just a note. When I tried getting rid of the 1 KB chunk process, the whole script blew up telling me the data was too large to read. So I added it back in. I'm not sure if the "too big" came from BIDS or the website, but I now know that I need it in this specific instance.

EDIT: Thanks everyone for all your assistance. It was very valuable.