How do I open a Mann-Co Supply Crate?
Solution 1:
It looks like you may have to shell out for the key if you want to open the crate.
This is from the TF2 wiki about the Supply Crate Key: "They are only obtainable from the store so far and there are no plans to make them available via the Item drop system."
As per the October 20th update the key to open a supply crate was made tradable.
Source: TF2 Wiki
Solution 2:
There is! You can trade for this key with other players.
Keys were not tradable initially, but October 20 patch changed that, so now you can exchange something for the key.