I plan to build a new order form for my billing and as each field of a form gets selected, I want the other fields to update accordingly.

Example:

Drop Down Menu 1: Product A
Product B

if product B is slected, the next dropdowns are changed to specific values relative to that selection.

Is this a php solution or would I need to integrate javascript into it too?

You could do it either way, but I would suggest using strictly PHP. There's no guarantees that your users will have JS turned on when they use your order form. Using PHP, even though it would require a page reload, would ensure that all users can use the form.