Wow Dave, that's a bit harsh!
I mean, I'm sure the Excel sheets aren't the instructions for how to do the transplant... don't they have people that already know how to do that? Surgeons?

An RFI, if I'm not mistaken, is a Request for Information. JB's probably sending out a questionnaire or somesuch thing about stuff related to some aspect of their transplant work. You know like:
"Dear Dr whoever, I understand you do "Pediatric Autologous" transplants. Would you be interested in viewing our range of scalpels designed especially for "Pediatric Autologous" transplants? (see attached Excel sheet for prices) Or do you prefer to use a spoon?"

So, JonnyBravo, provided no-one's going to die because the VBA code is wrong (what a strange healthcare system that would be) here's an idea:
[VBA]Sub SomeSubName()

'In VBE>Tools>References, add a reference to the
'Microsoft Excel Object Library

Dim xlApp As Excel.Application
Dim xlWB As Excel.Workbook
Dim xlSht As Excel.Worksheet
Dim para As Paragraph
Dim blnDeleteThis As Boolean

Const strXLPath As String = "C:\RFI\Excel Sheets\Organ transplants.xls"

Set xlApp = New Excel.Application
Set xlWB = xlApp.Workbooks.Open(strXLPath)

'loop through the shhets and delete as appropriate
For Each xlSht In xlWB.Worksheets
blnDeleteThis = True
For Each para In ThisDocument.Bookmarks("bmkTransplantTypes").Range.Paragraphs
If para.Range.Text = xlSht.Name Then
blnDeleteThis = False
Exit For
End If
Next
If blnDeleteThis Then xlSht.Delete
Next

Set xlApp = Nothing
Set xlWB = Nothing
'show the save as dialog
Application.Dialogs(xlDialogSaveAs).Show

End Sub[/VBA]I've assumed that each entry in the bookmarked text is it's own paragraph, maybe its comma delimited, in which case, you can Split the string into an array and loop through that.
Depending on how that text is formatted in Word, you may also have to trim off leading and trailing spaces and other non-printing characters to get a match with your worksheet name so a little testing will be required.
Let us know how you get on and let's hope the transplants continue without incident