AWS Device Farm has an API called ListArtifacts. http://docs.aws.amazon.com/devicefarm/latest/APIReference/API_ListArtifacts.html
This API will return a list of artifacts(files, screenshots, and logs). Each artifact will have a URL so you can download the file. Each artifact also contains a type, so you can iterate through the list of artifacts and find the ones where type is "VIDEO".
Caveat: There is a difference between the "type" parameter in the ListArtifacts request and the "type" property returned in the Artifact object. The type in ListArtifacts request only allows three values: FILE, LOG, SCREENSHOT. However, the type property in the Artifact object has several possible values that are documented here: http://docs.aws.amazon.com/devicefarm/latest/APIReference/API_Artifact.html