I'm in the UK, and we don't have poison ivy, oak nor sumach, so I've never actually seen them. However, the small leaved plant covering the ground in between the Hosta leaves and the one with palmate leaves appears to be Vinca minor, common name lesser periwinkle, usually has blue flowers in spring. The palmate leaved plant looks like Parthenocissus, or Virginia Creeper; there also seem to be what looks like random blades of grass. Images of Virginia Creeper and Vinca minor below
I found the link below; according to that none of these plants is poison ivy nor poison oak, but as I said, never having seen the plants in real life, I'm reliant on internet info only, hopefully someone who's actually seen the poison plants and been able to identify them with 100% accuracy will confirm. It seems the gardener you've employed is either unable to recognise them, or is hoping to charge more for dealing with 'poison plants'.