Looks like the registration process opened online on 11th April. There is an official website Shri Amarnath ji shrine board, kindly check. There are two ways one is offline - you have to visit authorised banks, ask for Yatra permit to Amarnath, fill the form and submit with documents, which bank people will guide you. Other way is online- you will have to click the above mentioned website and attend to required procedure. There is a fee to be paid and compulsory health certificate with photos have to be submitted.