Is there any way to get hands-on experience with Web Application testing while not working as a tester? Are there any initiatives where people post their production and real-world apps for people to test them?
If you're talking about "real-world" applications to test, then like schroeder mentioned, bug bounties are your best bet. But then, while companies have them, there are a few online platforms where you can find a bunch. Examples -
You register with these platforms and more or less just start pentesting live websites like Uber, Netflix etc (make sure to follow their terms and conditions!). These are just the tip of the iceberg. Many companies provide their own bug bounty programs, competitions and so on. Beyond these, well, you could always install instances of buggy applications (WebGoat etc) and practice more.