For a living trust you can use the Settlors SSN since it is a disregarded entity for tax purposes. Once the trust becomes irrevocable (eg; the Settlor passes away), you need to apply for a Tax Identification Number with the IRS. This is a very simple and quick process using the link below from the IRS website. Note you might see TIN and EIN used interchangeably.
https://www.irs.gov/forms-pubs/about-form-w-9