The Tucks kidnap Winnie because she's seen their immortality-granting water source in the woods.
In Chapter 5, Winnie sees Jesse Tuck drink from the spurt of water in the forest that he tries to keep hidden. It is this secret source of water—which we find out later gives eternal life to whoever drinks it—that nobody outside of the Tuck family was supposed to see, but Winnie saw it. When Mae Tuck walks up to Winnie and Jesse near the end of Chapter 5 and realizes what has happened, she places her hand over her heart, her expression becomes "bleak," and she tells her family: "The worst is happening at last." She means someone — Winnie — found out her family's secret.
As Chapter 6 opens, the Tuck family kidnaps Winnie. They're desperate to protect their secret, so they whisk her off her feet onto the horse, carrying her deeper into the forest.
Winnie, who doesn't even realize that the spurt of water was such a big deal, finds the experience very strange, especially considering how her kidnappers are now pleading with her not to be scared and promising they'll soon explain why they're taking her. She'd always imagined instead that if she were to be kidnapped, it would be by rough, ugly people, and she would be the one pleading.