They definitely haven't admitted that we provided them, but it's the only way he could have gotten them. It's like when the Israelis say they don't have nukes, or have only 5-10. He didn't have chemical weapons factories during the war (frankly, if he had, they would have been captured by the Iranians). He needed those weapons to repulse Iranian human wave assaults which were costly but effective. The WMD's really did basically save Saddam from being completely over-run by the Ayatollah. Iranian troops were only a few miles from Baghdad at one point, and that was with heavy machine guns, heavy armor, lots of artillery, lots of mortars, even more machine guns, and Iraqi Air Superiority in both fixed and rotary wing aircraft.

The war ended in 1988, and we had been a strong friend to Iraq through the war, and were the ones shipping him those WMD's throughout to fuck with the Iranians. The war bankrupted him though, and it looked like the Saudis, Kuwaitis, Qataris were all using his weakness to try and isolate his influence out of the Arab League and tear the country apart. The Saudis specifically smelled blood in the water. Having already lost a swathe of his oil refineries, he really needed oil exports to get moving again. When he realized the Kuwaitis were actually drilling into an oil well that was mostly in his territory, he flipped his shit. When he found out that the Saudis were helping, he knew he was being intentionally destroyed, all to the great amusement of Iran. He decided he was going to stop the Kuwaiti oil drilling (because otherwise his main national income would fucking die) and he made sure to ask the US State Department whether or not the Americans would have any objection to the war.

Infamously, the US SoS gave him a completely non-committal answer. Something along the lines of "the US would see an Iraqi incursion as undesirable, but as a completely internal Arab matter among the Arab states" (something like that). Saddam actually thought he had the blessing of the US to invade. Then, all the sudden, after a quick victory, the same people who had been selling him WMD's less than 4 years ago were now calling him an insane megalomaniac, hell bent on world conquest. Lo and behold, the fucking Saudis were asking for American troops, and the royal family of Kuwait was making up stories of his troops killing babies on ventilators in front of congress. Then, the former CIA director who was now president, was warning of Saddam's chemical weapons threat from his SCUD missile program. Dude got played: his own drug dealer was moonlighting as a cop.

As for the weapons themselves, put it like this, the Iranians themselves are still using American weapon stockpiles provided during the Shah. Not necessarily fighters, but small arms and the like that can be maintained and won't really go out of style. He had four years to develop an independent chemical weapon program from when the US cut off theoretically could have been. We don't know how long after the war ended that the US continued to sell weapons. And, maybe he did try to develop his own chemical weapons independent of US ... "investment"... but, he would still have needed the knowledge of how to make those from US agents familiar with the matter.

Our fingerprints are all over the Iraqi WMD program, one way or another.