Can you Gust Vayne when she enters a battlefield?

Generally no. When Vayne moves to an occupied battlefield with no items on the chain, combat is immediately staged and she gains the Attacker designation before either player has priority, making her 5 Might (outside Gust's 3 or less range). However, there is an edge case: if another unit with a triggered ability (like Traveling Merchant) moves to the battlefield first, its trigger goes on the chain. This creates a window where Vayne is at the battlefield but Attacker/Defender designations haven't been assigned yet (because the Showdown cannot open while the chain is pending). In this scenario, you can Gust Vayne as a Reaction while she's still 2 Might, before the chain resolves and the Showdown opens.