I am trying to recall an animiated movie that we watched in the past ten years. The main characters were a grandfather that looked like a sea captain and his granddaughter. The granddaughter had a dog and I recall the dog running off into the woods and the search that ensued. The dog may have been called Heidi, but not positive on that one.

I believe you are talking about "The Seventh Brother". The dog runs away in a storm, and lives with a family of bunnies. Is that the one?