Such a strange post on a lot of levels. First off, if you go to a company of any size and say, "I want to use my time with you to learn x", then that company will say, "Thank you. We'll call you if there's such an opportunity." You will, of course, never hear from them again.
Second, you clearly do not know how "the game" is played as a front end developer. You've done with the least with the least here by posting these few paragraphs. If you want to get someone to pay attention to your front end skills, then you need to show them your work. Period. And preferably over a long period you will show a history of excellence. People will see that and viola - you will get a call/email.
Third, your post comes across as either 'This guy is very humble' or 'This guy is a complete newbie'. The problem is that I can't tell which. That's a big problem for you. No one is going to take the time to ask you, "Hey, which is it - are you a guru or a noob?" thus you'll receive no contacts from such ambigiousness.