Originally Posted by Elizellen

Nope - cups are liquid (volume) measurements not weight measurements so you cannot convert cups to weight ounces, you have to find out how many ounces of each veg would fit into a measuring cup so it is easier to do if you buy a set of measuring cups which most supermarkets and cooking shops sell.

One cup = 8 USA fluid ounces (a little less than 8 UK fluid ounces I think) = 237 mls
So if you have a measuring jug you can use that, and fill it up to the 237 mls mark with chopped veggies to see how much equals one cup.
For stuff you don't want to chop up you can use a big measuring jug and put 237mls of water in it then drop your veg into it and keep adding more till the water level reaches the 474 mls mark.

Or if you have kitchen scales you can go to the USDA database and search for the vegetable you want and it will tell you how many grams a cup of it weighs.
http://www.nal.usda.gov/fnic/foodcomp/search/
Or download it to use when offline here http://www.ars.usda.gov/Services/docs.htm?docid=5720

BTW for converting lots of measurements including temperatures, weights (mass) and volumes there is a great free software you can download here.
http://joshmadison.com/convert-for-windows/