A paper version will also be available - by calling 119 but not through a GP.
Both will be available from Monday, 17 May, when the ban on foreign travel is eased.
The NHS app is separate to the NHS Covid-19 app, which is used for contact tracing.
People can already use the NHS app to:
Request repeat prescriptions
Arrange appointments to see their doctor
View medical records
It can also show vaccine statuses, including for coronavirus, but currently this feature must be enabled by a GP before it appears on the app.
The new update will contain a separate feature to display coronavirus vaccine records, so the government said there should be no need to contact GPs.
The app will not show coronavirus test results, but the NHS plans to incorporate this in the future, the government website said.
It advised people to register to use the app at least two weeks before travelling.
A paper letter can be requested only at least five days after a second vaccine dose and can take five days to arrive.