How to make HikaShop act as plain ordering system

Instead of hiding the prices in CSS, you should not put prices in your products. That will display the text Free that you can change to something else by overriding the translation in the Languages tab of the configuration by editing your translation file.
In the same manner, you can change the text of the email to state that you will contact them later on with their quote.
Also, you will want to switch the option "Display 'add to cart' button for free products" in the Display tab of the config so that people can add the products to the cart.

Then, when an order is placed in HikaShop, you can just set a total price and notify the user so that he can proceed to the payment on his orders listing (with the option "Payment of orders not paid immediately" of the checkout tab of the config activated).

You should be able to do that with CSS:
.hikashop_cart_product_total_value, .hikashop_cart_product_total_title, .hikashop_cart_product_total_value, .hikashop_cart_subtotal_value, .hikashop_cart_subtotal_title, .hikashop_cart_coupon_title, .hikashop_cart_coupon_value, .hikashop_cart_shipping_title, .hikashop_cart_shipping_value, .hikashop_cart_tax_title, .hikashop_cart_tax_value, .hikashop_cart_total_title, .hikashop_cart_total_value{
visibility: hidden;
}