I don’t know any free no-code tool to do this, but if you know some quite basic Javascript and want to get into learning Firebase it would probably be your best bet for free, as long as you don’t have thousands of bookings a month
Have you checked the Powr plugins library? They have booking form and it seems they have free account version. I don’t know your exact needs, but maybe this one can work for you. https://www.powr.io/plugins/booking-form
Another service that is sort of free is ConvertCalculator, they have a few examples of booking or order forms