You must use the Replace function to do this:
Replace(Me.CompanyName, "'", "''")
Note, however, that your DCount function may not catch everything that you wish. In your case, assuming that you wish to catch a duplicate where the CompanyName AND the Address are the same IN THE SAME RECORD, you'd need to do this:
If DCount("OrganizationName", "tblOrganizationsMaster", "[OrganizationName]='" & Replace(Me.CompanyName,"'", "''") & "' AND [Org Address]='" & Me.Address & "'") > 0 Then
MsgBox "An organization with this address already exists.", vbExclamation, strAppName
End If