Am I sinful if I allow students who have run away from their schools to sit in the internet cafe and use it, knowing their situation, especially since one of them begs for money to come and play with it?

It is not permissible to host truant schoolchildren and allow them to use the internet and play games, due to the Shariah prohibitions involved. These include: aiding them in abandoning a religious duty, which is studying, as it may be obligatory for them by religious law or by their parents' command; the money they use may not be their own; and aiding them in sin in the event they beg to secure money for playing, in addition to the internet potentially leading them to illicit websites.