If you have any music editing software that lets you hook up a mic, like Ableton (I think that's the name?), you can plug your headphone jack into your mic jack (assuming modern computers still have these), and with some faffing can get things to record.
It's not usually the best option because it is time and labour intensive, and the quality may not be up to scratch depending on how you set the levels. Also, as said, it is kind of illegal if you are only streaming. I've used this approach in the past when music I want genuinely isn't available to purchase in my country, on the grounds that I'm not actually depriving anyone of money.