Currently there is no any API to access MyMaps programmatically.
I can see a feature request for this in the public issue tracker:
https://issuetracker.google.com/issues/35820262
It looks like Google is evaluating the feasibility to implement the API, however, no timeline provided at the moment.
Please star this feature request to express your interest and receive further updates.
UPDATE
As of April 2018 it looks like Google decided do not implement the API for Google MyMaps and marked the aforementioned feature request as Infeasible.