Keep an eye out for characters of any kind coming to your town. We took our toddler to see Bearemy the Build-A-Bear mascot twice before we went to WDW. She wouldn't go near him at all at first, but gradually warmed up to the point where we were able to get a picture with both of them in the frame.

Maybe take your child to Chuck-E-Cheese if there's one near you.
When we were at WDW DD would get all excited when she saw the characters from a distance. She'd wave frantically and yell, "Tigger! Tigger!" But then when we got close to Tigger she'd wail, "Noooo Tigger! Noooooo Tigger!" Again eventually we got to the point where we could get a picture that had both of them in it. I had to hold her, and it was good if there was a sibling or two in between her and the character.
And eventually she was willing to reach out and - quickly - touch a characters hand or head. But it took a couple of days to warm up to that point. The characters are well-trained to keep their distance from fearful little ones, and have a knack for getting the kids to warm up. They may turn their backs and peek over their shoulders, etc. Or extend a paw and pull it back if the toddler freaks out, or go farther if the toddler is interested. That kind of thing. Lots of teasing, all with body language.
Oddly enough, my now-8yo was only slightly fearful of the head characters (Mickey etc.) when she was 2, but was terrified of the face characters (Snow White, Cinderella.) Now she loves them all!