People can apply via a public ballot to honour the theoretical physicist, who died on March 14 aged 76.
Applicants need to give their birth date, which can be any day up to December 31, 2038.
This has caused speculation that time travel could be real and the Professor Hawking foundation said the possibility of time travel had not been disproven and could not be excluded.
A spokesman for the Stephen Hawking Foundation said: “We cannot exclude the possibility of time travel as it has not been disproven to our satisfaction.
“All things are possible until proven otherwise.
“But so far we have had applications from all around the world and we do mean round – there are no flat-Earthers here.”
Within 24 hours of the ballot opening, around 12,000 people from more than 50 countries had applied for tickets.
People from as far away as the US, China, Bolivia and the South Pacific have asked to attend the event to celebrate the life of Stephen Hawking.
Travel blogger IanVisits noted the application form allows people born from 2019 to 2038 to apply, theoretically allowing “people born in the future to attend the service”.
He wrote: “Look out for time travellers at the abbey.”
Thousands of people lined the streets of the professors home city of Cambridge for his funeral on 31 March.
The ballot for tickets to his memorial service closes at midnight on 15 May.
Stephen Hawking threw a party for time travellers on June 28, 2009.
The next day, he sent out the invitations.
He said in the documentary Into the Universe with Stephen Hawking: ”Copies of it will survive in one form or another for many thousands of years.
“Maybe one day, someone living in the future will find the information and use a wormhole time machine to come back to my party, proving that time travel will, one day, be possible.”
Sadly for the professor, nobody attended his party.
The physicist will be remembered for his pioneering work in studying black holes and describing the origin of the universe.
The memorial will be held on June 15.