Is it possible to sort a Spreadsheet in Excel using VBA
by more than 3 fields (which is the limit of the Excel
functionality)?

You would need to do it the same as you would in Excel.

Sort by the lower level sorts, and then resort by the
higher level sorts.

In the Help:

Search on 'Sort'
Select 'Sort rows based on the contents of two or more
columns'
Steps 3 and 4 explain how to sort with more than three
sort levels.