Hi,
there are several websites that offer forecast, but the problem is that it is useful when you want to have weather for your location. Because you cache the result, in other words, you would load it say once or twice an hour to update it.
If however, you wanted to give users the possibility to choose a zip code, then you cannot cache it, and in such case you would easily exceed the limit for queries that are available in the free plan. If you want to offer forecast for any location then it is likely not going to be possible for free.
Yahoo used to have a free API with no API key required, but about 2 months ago they changed it and it became very inaccurate, buggy and very very unreliable. I remember I once tried it when it was still working, but remember there are many bots also, which will very quickly exhaust your free api quota.