Zapper itself does not charge a fee for use of its platform. However, Zapper may expose you to fees required by integrated DeFi apps. For example, you will pay 0.3% fees if you trade on Uniswap through Zapper. That 0.3% fee goes to Uniswap, not to Zapper, and it is a fee you would have to pay for using Uniswap on any platform (including Uniswap itself), not just on Zapper.
Additionally, when you transact on Zapper you pay gas fees to have the network’s miners confirm your transactions. Gas fees range in cost depending on how the network you are transacting on is built, and how busy the network is when you transact. These fees are a normal part of using blockchains. None of the gas fees you pay are collected by Zapper.