This fee applies to all bookings no matter if it’s online or through a call centre. They are a low-cost carrier so they make money out of such fees.
This is made so the cost of the ticket seems cheap but actually isn’t.
For example a bag can cost more than the flight, it’s €30 per person if you don’t check-in online but at the airport and €15 for a call centre request, something which should be standard especially if you just have a question.
You almost have no choice to pay this fee.
From this Travel.SE question Is it possible to avoid Wizz Air's administration fee?:
WizzAir's website says you can book via travel agents or at the airport ticket counter. Also lists 10€ as the call center fee.
But being an LCC, they probably don't pay travel agent commissions, so a travel agent would likely charge you a handling fee.
The webpage you cite, mentions that there may be additional fees charged by the handing agent at airports as well. If you live at WizzAir's hubs, you would likely be dealing with WizzAir staff not a handling agent, but at destination airports, you will probably be dealing with handling agents.
