Yes, in two ways: as a draft before you submit, and as a copyable history afterwards.
Drafts (before you submit)
We autosave the form to your browser as you fill it in. Leave the page, come back, and your answers are still there. The "Save as draft" button just confirms what's already happening on every keystroke.
Caveats: drafts live in your browser, so a different computer or browser won't see them, and clearing your browser data wipes them. Once you actually submit, the draft cache is cleared.
Copying a past request
Once you've sent a request in a given category, every future request in that same category shows a Recent submissions box at the top of step 2 (after you've picked your specialists). Each prior project has a Copy button: click it and the form fills in with the old answers. Tweak whatever needs tweaking and send.
If you'd rather start fresh, there's a reset form link in the same box that clears the fields.