Yes it would. We couldn't have ordered the Astra Zeneca vaccine by ourselves if we had been in the EU as EU states couldn't order vaccines already part of the EU order. We would I think have had to wait for it to go through the EU regulators as well (which it hasn't actually done yet although it is due soon. EU countries can I believe order their own supply of vaccines which are not part of the bulk EU order (I saw somewhere that Hungary has done so.
Who said that the EU with their greater bargaining power is getting supplies earlier than us. The row is because Astra Zeneca are refusing to give the EU supplies ahead of us when we ordered them three months before the EU.
We wouldn't have been able to give the Astra Zeneca vaccines yet if we had been in the EU so we would have done far far less.