If you're talking about index mutual funds, then the author is just plain wrong. Any open-ended fund offering daily liquidity will trade, and therefore produce market impact, to meet its daily redemptions.
If you're only talking about ETFs, then this is technically correct. Besides the occasional index re-constitution, unlevered index ETFs don't do any trading. However it's definitely not true that there's no market impact. As the fund grows (or shrinks) the shares just don't magically appear in the portfolio. Somebody has to go out and buy (or sell) those shares, and like any trading volume, that creates market impact.
The mechanism that ETFs actually use is something called "Authorized Participants" (or APs for short). Basically market makers have the right to create or redeem shares in the ETF. To create new shares, they go out and buy all the stocks in the index, then hand a basket over to the ETF fund manager, who then hands back new shares of the equivalent value. And to destroy shares, the AP hand over shares in the ETF, and the fund manager hands back a basket of shares from the index.
If there's high demand for investors to own the ETF, that'll push up the ETF's stock price. As the price rises relative to the index value, APs will detect an arbitrage opportunity. They'll go out and buy the basket of stocks in the index at a cheaper price, then create new ETF shares at the richer price, and pocket the difference. Vice versa if there's demand from investors to exit the ETF.
The mechanism keeps the ETF price closely pegged to the index, because the further out of line it gets the more arbitrageur activity pushes it back in line. While also flexibly satisfying investors' specific demand for the ETF at any given time. Basically it delegates the role of trading from the fund manager, who usually doesn't have any special expertise in trading, to highly specialized trading firms and market makers.
However, as you can clearly see, market impact most definitely exists. If a flurry of investors rush to enter or exit an ETF, then a huge amount of trading has to be done to create or redeem the shares. Just because the APs create this trading impact, instead of the fund itself, is a distinction without a difference. The underlying stocks in the index are subject to market impact.