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Soreness on the eyelid could be an oncoming Chalazion. A Chalazion is a blocked eyelash gland. This causes soreness and will eventually cause a little bump on your eyelid. Most times you can apply warm, moist washcloth to the eyelid and this will help to bring the bump down and will help the soreness go as well. A stye is also another option.
As long as it is the eyelid itself and not the actual eyeball, you can continue to watch it and see if it improves on it's own. If you feel like your eyeball is sore, of course, you need to have someone check this for you.
No, a Chalazion (nor a Stye) would not cause a swollen, painful vein in your eyelid.
If you are having this type of problem, you will need to let someone take a look at this.