Remarks by European commissioner for internal market, Thierry Breton
They have been touting this since February and are continuing to stick with that goalpost as we approach May. Perhaps vaccine supply won't be an issue but the rollout certainly needs to gather pace as most countries in the EU are struggling to even achieve at least 10% of full vaccinations as of today.
For some context, Germany has administered 21.9 million vaccine doses as of 18/4 but only 6.6% (5.5 million) of its population have been fully vaccinated. France is also thereabouts with 17.0 million vaccine doses administered as of 17/4 but only 6.7% (4.5 million) of its population have been fully vaccinated.