The problem with NLP is not the algorithm (although that is an issue) the problem is the resources. There are some open source shallow parsing tools (that's all you would need to get intent) that you could use but parsing thousands or millions of tweets would cost a fortune in computer time.
On the other hand like you said not all tweets have hashtags and there is no promise they will be relevant.
Maybe you can use a mixture of keyword search to filter out a few possibilities (those with the highest keyword density) and then use a deeper data analysis to pick the top 1 or 2. This would keep computer resources at a minimum and you should be able to get relevant tweets.
