I have read there are bullets that pierce bullet-proof vests. If true are they legal or il? Do YOU have any weapon that requires them?WHY?

Armor piercing bullets are illegal.

Are they legal or illegal?---turns out it isn't that simple.  Here's a very good link:

https://www.quora.com/Is-it-legal-to-buy-armor-piercing-ammunition

I only have a .22 Colt single action pistol from when I was a teenager---looks like the old West Colt 45.  I doubt "armor piercing" rounds are available for it, although I have never checked.  And I haven't fired that weapon in over 40 years.

From what little I understand, in some of your states armour-piercing bullets are legal.
The reasons would vary, as you suggest. Organised crime and war would probably the main uses.

I own no weapons of any sort.
Even if in an emergency I was forced to improvise, I would probably be neither quick nor strong enough.
Unless one counts my mind. I use it defensively to avoid risk.
Where we live is remote and extremely private. The area is safe.
Where we visit violence is unknown.

No.  
For starters no steel or hardened metal penetrator is allowed in any kind of handgun cartridge.

Rifle ammunition is allowed to have a penetrator but rounds considered armour piercing are not legal.  However nearly any centerfire rifle round is capable of defeating a kevlar vest.   Armour piercing in rifle ammo means it is designed to defeat steal and ceramic plates. Bullet proof vests and clothing we undertand is only meant to take on typical handgun rounds.
Certain rifle rounds with steel penetrators got into a grey area when some novel gun makers began to make AR-15 and other rifle patterns into pistol variations.   The ATF made specific exemptions for certain specific rifle rounds that contain steel penetrators when this came about as the cheap surplus ammo used at ranges in  5.56 NATO and .30-06 has them and the rounds with or without can defeat kevlar vest with or without them and aren't much more effective on plate either.

  No vest is truly and completely bullet-proof, they are merely bullet-resistant, and even then, there are differing types of vests that are rated for the levels of protection they offer.  The ratings system varies for the task or purpose of the vest, ranging from personal protection against light threats (celebs might wear one of these, for instance), everyday law enforcement uses, special law enforcement uses (riot control, active shooter, bomb squad, for instance), and military/combat uses, the last one requiring more overall body protection and include protection from heavier explosives, shrapnel, etc. 

  The caliber of ammunition being fired, the type of weapon being used, the distance between the weapon and the vest, the angle of impact, the location of impact, etc are all factors in determining whether or not the round (projectile) will penetrate the vest. Of course, the higher the caliber, the more likelihood of penetration, as is the more firepower in the weapon, the closer the distance, and so on. 

  Another consideration is that the round doesn't have to fully penetrate the vest in order to incapacitate, injure, maim, or even kill the wearer. The impact force of being shot can be so strong that the vest merely slows things down or absorbs a large percentage of the effectiveness, but there's no magical guarantee that the vest provides of completely escaping harm.

  Bulletin-resistant vests, being garments, feature sections that are sewn together, and are filled with thick padding, or plated panels, or a combination of both. Some of the padding or panels overlap at certain points, and there are other places on the vest that have no padding/plates st all, such as the seams connecting two parts of the vest. Theses gaps in protection are vulnerable points where rounds of ammunition can strike the wearer and have absolutely no bullet protection whatsoever. In instances such as these, even the smallest round fired from a weaker weapon from a long distance has the probability of causing the wearer damage or death.

  One startling revelation about these vests is that even though they are bullet-resistant, many of them can be defeated by edged weapons, such as knives.

  In conclusion, the ammunition itself may be legal or illegal; both can potentislly penetrate the vest depending on the conditions at hand. Does such illegal ammunition exist? Yes. I do not own any firearms whatsoever.
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