word2vec: user-level, document-level embeddings with pre-trained model
You are on the right track with averaging the word vectors in a tweet to get a "tweet vector" and then averaging the tweet vectors for each user to get a "user vector". Whether these average vectors will be useful or not depends on your learning task. Hard to say if this average method will work or not without trying since it depends on how diverse is your data set in terms of variation between the words used in tweets by each user.
I am currently developing a Twitter content-based recommender system and have a word2vec model pre-trained on 400 million tweets.
How would I go about using those word embeddings to create a document/tweet-level embedding and then get the user embedding based on the tweets they had posted? 
I was initially intending on averaging those words in a tweet that had a word vector representation and then averaging the document/tweet vectors to get a user vector but I wasn't sure if this was optimal or even correct. Any help is much appreciated.