We used a spiral bound book of 4x6 index cards & a click top Sharpie (bought both at CVS). It was easy to carry these two small items in our small backpack & just flip the card for the next signature. No dealing with a Sharpie cap either. We got a multi-color pack of Sharpies, using a different color each day,but the red was the best. Some of the lighter colors didn't show up well.
When we got home I made a scrapbook with the signed index card on one page & a picture of DD & the character on the other page. If you don't need on signature for each child, just do one & a picture of all kids with the character. Would they share one scrapbook, maybe making it family community property? You could do one for each child, but it would just be more time consuming!