There are no mail order companies in Shanghai that are pertinent for expats. It is actually popular with the local population, however, any that exist are only in Chinese.
Shopping online is gaining popularity in Shanghai. For expats, sites like eBay are largely inaccessible because they are in Chinese and accessing the UK or USA sites from China probably will raise suspicions of fraud, making purchasing difficult and selling almost impossible. Some bookstores and supermarkets that cater mainly to foreign tastes may offer delivery services but it is best to make an online or telephone enquiry to determine which these are.
A favorite site among Chinese people is http://www.taobao.com, where you can buy new and second-hand goods, as well as trade items. There is also a Chinese eBay (http://www.ebay.cn), but both of these sites are in Chinese only. There is a lack of China-based English-language websites, but more are appearing. Some international sites will deliver to this country, but the terms and conditions will vary, and you will probably need to pay with a foreign credit card and assume high transport fees.