This is Vivian. She went to Disney World for the first time this year. This little sweetie loves a princess, and would spend hours on end with any of them she encountered. The fur characters, the ones with their faces covered, not so much. Viv had breakfast with her family at Crystal Palace, and Winnie the Pooh bent down behind her, and tapped her on the shoulder. "Scared the petals right off of her." (If you don't know where that quote came from, we need a little more Disney Channel therapy.)
It didn't go well. Safe in her daddy's lap, she gave Tigger the stink eye, and dared him to come near her when he bounced around the dining room. Poor Eeyore didn't even try it. Piglet was a bit better received, because he's smaller. And he's pink.
I've talked to many families who are waiting out a trip to Disney because their children are afraid of characters whose faces are covered. I understand that, but you can still have a wonderful time at Disney, and avoid them. But, as a parent who's planned this fabulous vacation, you want that picture of your child with her (if not your) favorite character.
If you do want to try for the meet and greet pictures, be sure to tell the handler with the character that your child is a bit unsure about meeting the character. They will certainly accommodate you, waving from a distance, and even setting up a picture like the one here of Vivian. The photo pass photographer set up this shot, and I just love how Donald posed just like Viv in the background.
I would want to help my child overcome her fear. You can show your children the characters in a parade, and present it to them like it's one step up from seeing them on tv. At a safe distance and in your lap, let them wave to their favorite characters. You can also show them other children meeting the characters in the meet and greet lines. Maybe by the end of your trip, your child will be more comfortable with characters. That's something to consider when scheduling your dining reservations. If you have a little one that is a bit wary of fur characters, plan your character meals with them closer to the end of your trip. Vivian's breakfast with Pooh was the first character meal other than lunch with the princesses at Akershus. But, by the end of her vacation, she was more comfortable with Mickey and Minnie when she met them at Chef Mickey's.
Overall, don't let "Character Anxiety" keep you from visiting Walt Disney World. The look on their faces when they see the castle for the first time far outweighs a little nervousness about Mickey.