Make the marinade ahead of time; freeze it and stash it in your cooler to help keep everything else cold.
Clean, trim, and peel vegetables. Slice the larger ones (such as squash and eggplant), into ¼-inch-thick pieces. Skewer small vegetables and onions. Place in two extra-large Ziploc freezer bags. Combine remaining ingredients and divide between the two bags. Refrigerate until ready to use.
Heat grill to medium low. Shake off excess marinade to avoid flare-ups. Stagger vegetables by size and density. Grill whole beets 30 minutes (or parboil 20 minutes and grill 15 minutes). Grill carrots, fennel, peppers, and onions 20 to 25 minutes. Grill eggplant slices 10 minutes. Grill zucchini, summer squash, portobello mushrooms 5 to 7 minutes.