Nevermind, I was able to update this on my own by modifying the code that macropod provided and using Application.GetOpenFileName. I realize this isn't pretty and the while loop is redundant for the single file, but hey, it works! I have this as part of a userform, hence the private sub. Here it is:
Private Sub File_Button_Click()
Application.ScreenUpdating = False
Dim wdApp As New Word.Application
Dim wdDoc As Word.Document
Dim CCtrl As Word.ContentControl
Dim strFile As String
Dim WkSht As Worksheet, i As Long, j As Long
strFile = Application.GetOpenFilename
If strFile = "" Or strFile = "False" Then Exit Sub
Set WkSht = ActiveSheet
i = WkSht.Cells(WkSht.Rows.Count, 1).End(xlUp).Row
While strFile <> ""
i = i + 1
Set wdDoc = wdApp.Documents.Open(Filename:=strFile, AddToRecentFiles:=False, Visible:=False)
With wdDoc
j = 0
For Each CCtrl In .ContentControls
j = j + 1
WkSht.Cells(i, j) = CCtrl.Range.Text
Next
End With
wdDoc.Close SaveChanges:=False
strFile = ""
Wend
wdApp.Quit
Set wdDoc = Nothing: Set wdApp = Nothing: Set WkSht = Nothing 'T
Application.ScreenUpdating = True
Call Cancel_Click
End Sub