Warm up your shin muscles before you start walking. Think of the vigorous walking that is a Disney vacation as any other exercise. Don't hit the ground running without warming up your muscles. If you are going to stretch, warm up before you stretch too.
To warm up your shins, do simple toe taps. One foot at a time. It's easier to do if you have both knees bent, with the majority of your weight on the foot that isn't doing the toe taps. Begin with a simple up and down. Then switch to tapping in and out (toward the other foot and away). Return to the simple up and down. Repeat on the other side. Repeat as necessary during the day if you feel yourself stiffening up after sitting for a long time (a long meal or show, for example).
Stretch at the end of the day for sure. If you drive, also try using a "roller of pain" as we call it. This is a hard foam roller that is the equivalent of a rolling pin for your muscles. It doesn't feel very great while you are using it but it can seriously help if you develop knots.