I'm actually looking to become a set designer for Lego or a product designer for Hasbro, too, and I'm currently at my third year in university.
Pardon my english, I hope you can understand the following sentence, as I don't know how the education system works in your country, but only in mine, Italy.
If I remember correctly, to get such jobs you should get a three years degree in Product Design, in the case of Hasbro you can also get an Engineering degree (3 or 5 years, I don't really remember).
After that you'll have to create your portfolio, which is a "collection" of your projects made in university, in work stages, etcetera, which you will present to your job interviewer.
As for contacts, if you cannot find contacts or that infamous "work with us" on thir sites, you could try and look up for their "sister locations" and ask them. Even going to expos could be a good idea to try contact Lego or Hasbro.
With that said I wish you best of luck, hope to meet you when we will both get the jobs of our dreams.