When Shadow is played on a battlefield, does it open a window to react?

Yes — playing Shadow opens a chain, so opponents get a Reaction window before Shadow resolves.

Any card being played becomes a pending item on the chain, which puts the turn into a Closed State; only Reaction-speed cards and abilities can be added at that point (Rule 402.1). So your opponent can respond with something like a Reaction spell or a Hidden card revealed from facedown, and if they remove Shadow while it's still on the chain it never reaches the battlefield at all.

One clarification: the window isn't caused by any "when you play me" trigger — Shadow has none. "If you play me to a battlefield, I enter ready" is a replacement effect on how it enters (Rule 369.3), so the only chain item is the play itself. Reminder that Shadow's [Action] is timing permission only (showdowns, any player's turn) — it doesn't make Shadow itself playable during someone else's Closed State.