First of all, Aura in RWBY is not entirely consistent through the show. In Volumes 1-3 the rules are not fully established and it is guided more by the rule of cool to let the characters being able to tank more damage. From Volume 4 it is visual better represented and has more consistent power scaling. The basic concept however hasn't changed.
Everyone with a soul has Aura. A basic introduction into the concept was given in the fourth episode of the supplementary series "RWBY: World of Remnant" (Remnant is the name of the world in RWBY), which was released on the RWBY Volume 2 DVD and later on youtube. A transcript is available on the rwby fandom wiki. It is explained that
Aura is a manifestation of the soul, a life force that runs through every living creature on Remnant, whether they are a meager shopkeep or a renowned knight. However, what sets true warriors apart from all others is their ability to amplify and control their Aura. Aura is primarily used as a defensive mechanism. Passively coating the wielder in a protective force field, it can protect a combatant from what would normally be a fatal blow. It does not, however, make the user invincible.
So the main "practical purpose" of Aura is defence. Another introduction into the concept is given in Episode 6 of Volume 1 ("The Emerald Forest"). 805
<PERSON> explains:
With practice, our Aura can be our shield. Everyone has it, even animals.
She goes on to "unlock" <PERSON>'s Aura.
This is picked up again in Episode 4, Volume 5 ("Lighting the Fire"). <PERSON> and <PERSON> are training hand-to-hand combat and after <PERSON> is knocked down by <PERSON>, <PERSON> (in Oscars mind) says:
You forgot to engage your Aura, again.
Shortly after <PERSON> explains to <PERSON>:
Your body isn't used to this kind of training, not to mention generating a defensive Aura on your own. It takes intense concentration at first, but in time it will become second nature, allowing you to deflect attacks and gradually heal your wounds.
So even when someone has unlocked their aura and knows how to use Aura defensively, it needs actively be engaged. However, we as viewers have usually have no indication that Aura is engaged except when someone takes a heavy blow or Aura is exhausted and the Aura is flickering.
Since Semblance is Aura in a more tangible form, we know everyone with a Semblance has unlocked and trained Aura. The three most prominent characters that don't have a Semblance are <PERSON>, <PERSON> and <PERSON>. <PERSON>'s Aura breaks in the fight vs. <PERSON> (Episode 6, Volume 3, "Fall") and <PERSON>'s Aura saves him from a bullet in his fight vs. <PERSON> (Episode 11, Volume 7, "Gravity"). We have never seen <PERSON>'s Aura break or flicker, but his fights in Volumes 1-2 happened prior to the introduction of visual clues for Aura breaking. Only his last fight, in which he was swallowed whole, happened after the introduction of those visual clues. I would assume that he uses Aura: in his fight with <PERSON> in Episode 11, Volume 2, "No Brakes" he takes several sword hits mid-air without injury and continues to fight after. Furthermore it was stated by the RWBY creators that he originally was going to have a Semblance, but they never had the chance to include it (Director's Commentary Volume 7, Episode 7).
If you consider the likes of <PERSON> as a main character, then "non-warrior" characters like him are your best bet for someone not being able to use their Aura, it is never explicitly stated however. 663