I like to hunt at Goodwill, but you may have just the thing stashed in your closet. The ideal sweater will be a fine gauge (ie small stitches of lightweight yarn) wool (preferably 100% merino, 100% cashmere, or 100% alpaca) with sleeves that are joined to the body of the sweater in a diagonal line from the armpit to the neckband. Also, the sleeves of the sweater should ideally be the length of your child’s leg, with their crotch meeting the armpit of the sweater. I used a woman's small, and the sleeves were *just* long enough for my 4-year-old’s legs. If it feels scratchy to the inside of your wrist, it will probably not be comfortable on your child--find something softer.
May I rhapsodize about the virtues of wool for a moment? It is a remarkable fiber--able to keep you warm even when it is wet, it does not burn, it is extremely hard-wearing, it repels dirt, and it (paradoxically) keeps you cool. I know many people think it itches--it certainly can!--but there are some unbelievably soft wool fibers out there. The one everyone has heard of is cashmere, but merino and alpaca are close seconds. There is often at least one cashmere sweater in the racks at Goodwill when I visit, and I can always find merino or merino/cashmere blends. I live in a smaller city, so take heart and go hunting!