Vowpal Wabbit and LSH

No.

In LSH, you hash items so that nearby items are probably are close when hashed. LSH is a way of doing approximate nearest neighbors quickly. LSH uses a special kind of hash function.

With VW, you hash features so that you can reasonably support an unknown a priori and changing feature set, like you'd have with natural language when using words as features. In batch style learning, the typical way to handle a bag of words representation is to build a structure that maps words features into indicies. But it has two disadvantages compared to the hashing trick, you need to keep that string -> integer index mapping around which takes up significant amounts of space, and you can't really add new features online, since you tend to have a fixed sized feature vector, with one weight per feature.

The key insight with the hashing trick is ignore the perfect word -> index mapping, your learning algorithm will optimize around collisions introduced by the hashing function, especially when you have high dimensional and redundant training instances. The hashing trick uses a standard hashing function, and is mostly about having the bravado (and some combination of proofs and practical experience) to be okay with allowing some random set of features to share a common weight.