Hi,
One solution is to set no price for your products.
It will then display "Free" in the product pages but you can override the translation to replace the text "Free" By "".
You will also have to go in Configuration > Display and set "Yes" for the option "Display 'add to cart' button for free products".
Finally, for the cart module, disable the option "Display Price".
You can also override the view "checkout / cart" to remove the price column.
Hope this solution will be ok
In Configuration > Checkout, remove the steps "Shipping" and "Payment", and you will receive only the orders.