Would the income from a job be unlawful if the work uses stolen electricity, given that the questioner is unemployed and wishes to engage in e-commerce?

If your work is permissible, then your earnings from it are lawful and are not rendered unlawful due to electricity theft. However, theft itself is forbidden, and you must cease it and repent from it by returning the value of what was stolen to its owners or seeking their forgiveness. If that is not possible, then donate its value to the poor or to public welfare projects.