Hello itconsulenza,
First of all thank you very much for using the WP Travel Plugin and Travel log theme.
The booking form is only displayed may be because you have not enabled any payment option from Admin Panel > Appearance > Settings > Payment.
Also, you need to make sure to assign the cart page and checkout page by going to Admin Panel > Trips > Settings > General.
Now to guide you through this and help you set up the cart, checkout page please refer to below given link of our official documentation.
http://wptravel.io/documentations/user-documentation/settings/general/
If you find any difficulty/confusion further then please let us know.
Thank you.
Best regards!!
Have a good day 🙂 .