If you're committed to avoiding chemicals in your ivy removal, lots and lots of hand pulling is the main option. If you are starting in on a big ivy removal project, first create ivy-free circles around the bases of trees to protect them from being overgrown and toppled by a heavy growth of ivy. Also prioritize removing ivy that is flowering or fruiting, and keeping the ivy contained in the area where you first found it. And of course since ivy re-sprouts from tiny fragments, please do not mulch or try to compost the ivy you remove, but bag it and dispose of it in the trash. If you clear an area, keep checking for new sprouts and pulling them out over time to ensure the ivy doesn't reclaim the ground you have gained.
If you become overwhelmed and end up contemplating use of chemicals after all, cutting the ivy and then spot treating the freshly cut stump can be quite effective at killing ivy with minimal use of chemicals (relative to spraying the surfaces of leaves).
More details about English Ivy are available in the PNW Weed Handbook. Of course always follow directions and precautions on the label.
I hope that helps you get started and good luck!