Vaccines go into muscle, not blood vessels.

It's still much better for everyone involved to not stick air into your muscle, though.

Any air accidentally injected into a vein as part of a vaccination would be insufficient to cause injury. It takes tens or hundred of milliliters of air to cause a venous embolism in a typical person.

Covid vaccine syringes are 1 ml. It's impossible to inject enough air using one of these syringes to cause any sort of trouble.