ESTA applications may be submitted at any time prior to travel. The processing of an ESTA can take up to 72 hours.
Traveling with an EAD Card: It is generally not permissible for individuals to travel abroad on an EAD card alone. Depending on your immigration status, you will need additional travel documents, such as a passport or a refugee travel document.
If you are a TPS beneficiary applying for a new travel authorization document, you should continue to use Form I-131, Application for Travel Documents, Parole Documents, and Arrival/Departure Records. If you have a pending Form I-131, you do not need to file a new application.