When your vehicle passes the MOT test, the result is entered into the national database of MOT test results. The pass/fail data has already been available for a while via the MOT history lookup page, however you had to go to a service station to get a replacement certificate.
In 2019 DVSA has launched a new free online MOT certificate replacement service on the Gov UK website that you can use from the comfort of your home which is even more relevant in 2021.
You can request an MOT certificate replacement at any MOT testing station (it does not need to be the station that carried out the original MOT). Replacement at the MOT station is convenient if you need a printed copy and don't have access to a printer. The government regulates the certificate replacement fee which is capped at £10.
You can now get a free replacement MOT certificate online using your cars registration number and V5C number from the log book.