Aura of Vitality does not create a visible area of effect.
The description of the spell says:
Healing energy radiates from you in an aura with a 30-foot radius. Until the spell ends, the aura moves with you, centered on you. You can use a bonus action to cause one creature in the aura (including you) to regain 2d6 hit points.
The word radiates means simply that the source of the healing energy is the caster and it may affect creatures in a spheric area of 30ft radius: there is no indication that the aura is visible.
As already noted in Groody's answer, if the aura created by a spell produces a visible effect this is clearly stated: notably, Holy Aura seems to be the sole spell with this name that has a visible effect.
Therefore, if Aura of Vitality has been cast before the invisibility takes place (regardless how the caster became invisible), using the effect of this spell does not shed lights nor gives hints on the position of the caster.
If invisibility takes place before the casting of Aura of Vitality there may be some issues.
The question does not state how the caster has become invisible. Let's consider the 3 main cases.
Invisibility spell.
The text of Invisibility states (emphasis mine):
The spell ends for a target that attacks or casts a spell.
Hence, if the caster was invisible due to the effect of this 2nd level spell, casting Aura of Vitality ends the invisibility.
Greater Invisibility spell.
The text of Greater Invisibility states:
You or a creature you touch becomes invisible until the spell ends.
This time there is no ending due to spell casting: hence the caster of Aura of Vitality remains invisible. Anyway, the material component of Aura of Vitality is verbal: hence, the magic chanting may give hints to enemies where the caster is.
Invisible because hidden.
The caster is hidden because, for example, they took the Hide Action before casting, and hence they are unseen by enemies. Following the rules for hiding:
[...] you give away your position if you make noise, such as shouting a warning or knocking over a vase [...]
Since casting Aura of Vitality requires a verbal component, the caster is making noise, hence they are giving away their position.