We had the most success with using flax or chia, and water per the directions. Make sure the water is warm and really let it sit and get gummy. Egg Substitute just didn’t seem to be working. Pumpkin puree is a good idea for the drop cookies! Or try brownies, they tend to be more forgiving.
I think it has more to do with what you are trying to make. A cake is going to have trouble rising enough without eggs; a cookie or shortbread or something like a 7-layer bar is not going to mind quite as much. I’ve had a lot of success with recipes from a site called The Loopy Whisk. Good luck! (And thank you for trying – you’d be surprised how many grandparents won’t)