Dyeing polyester, in my limited experience, is absolutely thankless. You are braver than I am to attempt it. It is entirely possible, but keep in mind you can not dye polyester like you would cotton.
I-Dye poly does get the job done. However I've found it isn't very good for really dark, rich colors. I was trying to dye a light gray base to black and it ended up some really weird bluish grayish color. I wouldn't use sharpies, for the smell and the fact they have to be dissolved into alcohol which can damage/discolor fabric. In the case of felt I Would think it would make it kind of...sticky too.
You must use the stove-top dyeing method for polyester. You can read about it here
and see what you are getting yourself into. Make sure you have a pot big enough for the vest, and one big enough for the fabric. It's pretty straight forward once you get started but I had a horrible time finding containers big enough to dye in.