Hi,
You can set no price for the products, then it will display "Free" in the product page, override the translation to replace the text "Free" By "".
Then in Configuration > Display, set "Yes" for the option "Display 'add to cart' button for free products".
And for the cart module, disable the option "Display Price".
You can too override the view "checkout / cart" to remove the price column.
In Configuration > Checkout, remove the steps "Shipping" and "Payment", and you will receive only the orders.