The vaccine is usable, in the sense of being able to be used, rather than applicable.
I may be splitting hairs, but I think a vaccine is something which is used or administered rather than applied. If something is applicable, this means to me that it is relevant to a particular situation or problem. Or an adjective may or may not be applicable to a particular person. ("speedy" does not apply to me, whereas "pedantic" often does!). One might apply sticky tape to a package, but I don't like the idea of a vaccine being applied to me, although I shall want to be given a COVID-19 vaccine as soon I get the chance.
In this particular context, I would translate the phrase (... die Entwicklung von RNA-basierten Impfstoffen an , die binnen kürzester Zeit eine Anwendungsnähe erreicht hätten ...) somewhat loosely as follows:
... the development of usable RNA-based vaccines in a very short space of time ...
A more literal translation would be unwieldy in my view.

"As an example, the President cited the development of RNA-based vaccines, asserting that they had become a great success by reaching a (high) level of practicality within a short amount of time."

It really depends what he's referring to.
This is how I understand it (1.).

1.
"came within reach of usability" - Additionally to those vaccines on the market now, this would include all other promising developments, that were coming close to being approved, but in the end may not have made it.

2.
"became ready for application"
"gained product/market maturity" - These suggestions would be suitable for the approved and released products only.