So, why did Whitley send ED-E on a cross country journey across the USA? First Chicago, then Navarro, then to the Mojave. What was ED-E's purpose?

Sentimentality, primarily. Keeping ED-E from getting liquidated when Autumn ordered all work and resources on the Duraframe project towards Hellfire armor. Officially, to deliver information on the Duraframe Eyebot project, plus the only working prototype, to other Enclave outposts.

First Chicago, then Navarro. Mojave was where Ed-E unintentionallybroke down on its way.