It is not currently possible to access the Essential API using a user account which authorises via SAML however it is possible to create a dedicated local account (non-SAML) for API access.
In Essential Cloud, go to Configure > System Administration > Manage Users and create a user with the appropriate permissions that you will use for API access. In the email field you should enter an email address however this does not have to be an actual email account. This might be something like firstname.lastname@example.org.
Send an email to email@example.com with the user name that you wish to use with the API and the EAS support team will then provision and send you a temporary password for this account.
You should the log into Essential Cloud using this account and temporary password and change the password immediately using the User Settings page (top right of Essential Cloud).
You can then use this account for API access and manage its permissions like any other account.