No, it cannot stack with itself
Bonuses from the same source (feat, spell, class ability, racial ability, etc) will never stack with themselves, as clarified in the definition of Bonus:
Bonuses are numerical values that are added to checks and statistical scores. Most bonuses have a type, and as a general rule, bonuses of the same type are not cumulative (do not “stack”)—only the greater bonus granted applies.
The important aspect of bonus types is that two bonuses of the same type don’t generally stack. With the exception of dodge bonuses, most circumstance bonuses, and racial bonuses, only the better bonus of a given type works. Bonuses without a type always stack, unless they are from the same source.
So, using the same feat twice will not grant it's bonus twice. Similarly, using the spell Bull's Strength twice (even if two different casters do it) will not grant a total of +8 to Strength.
Unless said ability specifically says that it stacks, but those are rare. For example, the Fleet feat.
As for being strong or not, I would rather not suggest allowing an exception and trust (with a grain of salt) that the developer who wrote it to decide if the feat balanced or not. There are feats with similar effects that call that they stack, and a bunch of Aid Another(-ish) type feats that do not stack, and the benefit of this feat is granting an ally the benefits of another feat for one round (Lightning Reflexes) which seems balanced (to me) for a feat. Especially if you consider that despite the requirement, you are not actually required to use a heavy shield (it could be a tower shield), and you could grant the Vanguard Style bonus to multiple allies against a single area attack.
Multiple Aid Another actions from different allies will stack. While most of Paizo's community mostly agrees that one character still cannot make multiple Aid Another on the same person (evidenced by how it also didn't stack in D&D 3.5), it is unclear and should be discussed with your GM. Here, however, the source of the bonus is not the Aid Another action, but the Vanguard Style feat.
Similarly, a cavalier of the order of the dragon cannot use their Aid Allies multiple times on the same ally, because the ability is not increasing the bonuses from Aid Another, but replacing its effects by something else. However, it may be used together with other abilities that increase the bonus received from Aid Another, such as the Helpful trait.
If ruled that they should stack, there are several combinations of class abilities, feats, and traits that can increase the bonus from Aid Another, and if this mechanic is interpreted as presented here, you could spend 3 or 4 AoO attempts to get a +20 (or higher) to reflex saves on demand. See order of the dragon, Helpful, Pathfinder Chronicler, Inheritor etc. This wouldn't be that powerful if they also couldn't do the same to grant up to +14 to AC whenever necessary combining Benevolent, Bodyguard, and so many other abilities that can increase Aid Another's bonus to AC by spending attacks of opportunity out of their turn.
Vanguard Style alone is pretty bad
The feat, as written, has very similar wording to the Bodyguard feat, which, according to this FAQ on Bodyguard, still has the same limitations of the Aid Another action:
- The attacker must be within your threatened area;
- The ally must be adjacent.
This means that you will only be able to help on reflex saves if the attacker is very close to both of you, such as a dragon's breath or a point-blank fireball. Which is why it must be combined with Combat Patrol (which increases your threatened area) and reach weapons so it can be efficient. Using the next feat in the chain, Vanguard Hustle removes the attacker's restriction on the bonus as long as that ally is within your threatened area, but the attack must still be within your threatened area to use Bodyguard or Vanguard Style on another ally, or to activate it again on an ally that left the area.
While the author of Bodyguard, Jason Nelson, has clarified that he originally intended for Bodyguard to be used against any attackers, not only those who you threaten, he also said that he misremembered how those rules worked and it might have affected the wording of the feat. But even after he posted that clarified, the FAQ that came afterwards has clear on it requiring the attacker to be within the threatened area of the bodyguard character.
That said, if you are the GM, you may rule the feat however you want outside of PFS, which is the only scenario where GMs are required to rule according to the rules as written. That would give more power to the feat, sure, but the feat alone is pretty bad, as I said earlier, only getting better (due to the increased range and options) when applied with other feats. Ruling that it works on regardless of the location of the attacker simply means that one specific (and rare) use of Aid Another will get buffed, nothing else.