If Shelob is representative of spiders in general, it would not have matter WHERE she stung him. She could have easily penetrated the chainmail shirt and its underlying padding because of how spiders administer venom to their target.
Cross-section drawing of spider's fang and poison sac
The spider's fang would easily penetrate chainmail because chainmail is a layer of links with holes in them. Even if they were covered with tiny scales to offer some layer of overt protection, chainmail is vulnerable to piercing weapons because they can slip between the links.
Shelob's sting, if it works like conventional spiders should have found little or no difficulty penetrating the chain shirt worn because she would have grabbed her prey with her cheliceral teeth, pinning him still and then moving her stinging fangs around on the armor until she found a spot she could penetrate, even if the armor was resistant to her first probe, like a real spider she would simply reposition the prey until she found a spot. She would not have to damage the armor to slip just a bit between a less well-defended spot and release the toxin.
Underside view of a spider's fangs, tucked away for convenience.
Most spider fangs are relatively long and slender designed for deep penetration into insects who come equipped with their own natural armored chitin for protection. There really wouldn't have been any armor capable of resisting Shelob because of the nature of how armor is designed. For someone to be able to move, lift and wear armor for extended periods, there are always vulnerabilities to be exploited and chainmail is no exception, which is why it was often worn in combination with plate armors over more vital regions. Even then, it was not always proof against arrows, particularly the English longbow which in the hands of an experienced bowman could penetrate plate armor.
Early French combined chain and plate mail
Unfortunately the movie production of Shelob did not seem to recognize Shelob was supposed to be a spider, not a wasp, so she should not have had a stinger, she would have had fangs to deliver an envenomed bite. So let's assume either Tolkien didn't know how spiders delivered venom or there was a production mishap where no one knew, either. Or there is a third option, she only resembled a spider and was just a monster with the appearance of a spider. In which case we can't really make any assumptions about her abilities other than she is a formididable enemy who has survived centuries killing everything that had ever found its way into her lair until Samwise, Sting and the Phial of Galadriel injured her for a time.
In Peter Jackson's film trilogy, Shelob's appearance is held over until the middle of the third movie, The Lord of the Rings: The Return of the King. In the movie, Shelob can be seen to have a retractable venomous stinger at the rear end between the spinnerets, resembling a wasp's stinger, unlike real spiders which inject venom with their fangs. Shelob also appears to have a gaping mouth, whereas real spiders can ingest only liquid. In a DVD commentary, Jackson says Shelob's appearance is mostly based on the tunnel-web spiders of New Zealand, which he hates.