The primary issue is - whether they are entitled to object. Is stock market data Proprietary of the exchange, or is it public property. It is being generated by thousands of traders, who are buying and selling stocks. A company like Dotex or similar can process this data and provide it in a certain format for distribution etc. But can they lay claim to ownership of that data?
As for showing realtime charts on websites, the charts are the website owners creation and intellectual property. There are many people out there providing realtime data for charting purpose, both big and small e.g. esignal, thomson, and smaller players. Whether the website owner uses the data from these vendors for his personal use or commercial use, will be bound by the contract between the website owner and data provider. Dotex should not have a role to play in this.
Mining of data from trading terminals may be "illegal" technically. It's the same like using cracked software/pirated copies of Windows/ downloading music/movies etc. Finally, it all boils down to the use - commercial or personal, which prompts companies to go after people. Personal use is generally ignored by companies for various reasons. Otherwise, technically, these companies and sue almost 90% of web users for doing some kind of illegal activity. Such are our laws