The hard hats should be marked underneath somewhere with letters. Those letters will tell you what type of plastic they are molded in (ABS, TPO, PC+ABS, PP etc.) The type of plastic you are dealing with will determine how it is prepped. A common misconception with painting plastics (and I've seen it numerous times on this forum) is you need an adhesion promoter. This is not always the case. There are quite a few plastics that paint will adhere to very well without using an adhesion promoter (ABS is one example). It doesn't hurt to use an adhesion promoter if you wish to do so, but it isn't always required... Just thought I'd mention that.
Now for the bad news... Hard hats are mostly made from polyethylene plastic (will be marked PE). PE is pretty much impossible to make paint stick to it for extended periods of time. I painted a dozen Polaris Rangers for a client some years ago. The bedsides and tailgate on those little side-by-sides are molded from PE. These were all show vehicles and did not get used for the purpose they were designed for. Fortunately only one came back. They used a power washer to clean it before a show and the pressure from the washer peeled a huge section of paint right off the tailgate all the way down to the original plastic. I warned them not to use a power washer for that reason prior to this happening, but I still warrantied my work because it was a very good client. Point is, it can be done, but I would only paint PE if it's for show purposes only! If the hard hats are intended to be used on the job site the paint will come off.
As far as prepping PE... I've tried every method I'm aware of from heating the plastic with a blow torch to using House of Kolor's AP-02 (an adhesion promoter designed specifically for PE plastics). The method I found that yielded the best adhesion and durabilty was using Spies Hecker Permacron Elastic Primer Surfacer. Scrub the part clean with warm soapy (I prefer Dawn dishsoap) water, scuff the surface and apply the primer and color. I've had great success masking over this base for art work. Hope this helps and if you have any other questions feel free to ask. Good luck!