Is there a way to move items from quote back to basket?
No. You cannot put items from quote back to cart, but there is a better option.
As per help topic on Bricklink:

You can add quote items to a wanted list at any time by clicking the Add Quote Items to Wanted List button on the quote summary page

Here how it looks in your quote (just at the bottom of your item list):

Once you click on that button you have an option to add items from your quote to an existing Wanted List or create a new one. This then allows you to have a list of items from quote that you can use to buy in other stores too, not just the one you've requested a quote from.
More details on how to use wanted lists when buying parts can be found in another help article on Bricklink.