Yes, for non-fiction you don't have to write the book before pitching it direct to publishers.
You'll need to put together a book proposal which includes a covering letter that tells them about you and what qualifies you (what specialist and insider knowledge you have) to write this book - this is your 'author platform'.
Your target readership - the fact you've already had a request for a copy is good, as it signifies there is a gap in the market for this type of book.
A short synopsis or overview of the subject you will be writing about, and what areas you will be concentrating on.
A list of chapters and short chapter outlines/summaries - you might change these when you actually come to write the book, but it's simply to show the publisher that you know the subject matter and konw how to structure a non-fiction book (similar to structuring a dissertation).
Last but not least a sample of your writing, so a couple of pages from one of the chapters.
If they like the sound of it they will give you the nod to go ahead and write it. You will need to give them some idea of how long it will take.
I'm afraid I don't know if they pay an advance at that point, or pay the Advance on delivery of the mss.
Most non-fiction writers deal directly with the publisher, so you don't need an agent.
- NaomiM
<Added>If you google 'non-fiction book proposal' you'll find a number of templates online.
eg,
http://www.catalystonline.co.uk/GUIDELINE%20FOR%20A%20NONFICTION%20SYNOPSIS.pdf