Hi Alexis! Welcome to the Disney Parks Moms Panel! I love helping fellow Texans plan their best Walt Disney World Resort vacations ever!
Hollywood & Vine is located inside Disney's Hollywood Studios, so yes, you would need a valid theme park admission ticket to dine at this location. You do have a few options for
Character Dining not in the theme parks. Those would be: the Good Morning Breakfast with Goofy and His Pals; Cape May Cafe; Trattoria al Forno; Chef Mickey's; 1900 Park Fare; Disney's Perfectly Princess Tea; 'Ohana; and Mickey's Backyard BBQ.
Since you're planning a down day, which PS I highly recommend not only for children but any adults in your group, have you thought about what you'll do for the rest of the day? I would recommend visiting
Disney's Boardwalk as there are multiple fun things to do in the area to justify the trip, but it's small enough of an area to see everything in a few hours and then head back to your resort hotel for nap or to play in the pool. If you have more time you'll for sure want to check out
Disney Springs. It's the ultimate shopping, dining, and entertainment district that literally has something for everyone. Bowling, movies, dining, the Western Hemispheres largest Disney Store, even
driving a car in a lake!
I hope this helped your planning process, Alexis, and we'll see you real soon!
~Kristen