Can a classifier be trained with reinforcement learning without access to single classification results?
If every data point has a ground-truth label (i.e., one of the six labels), than any supervised learning technique can work, including Random Forest. If the labels come in batches, then the parameters of the model can be updated with each new batch. Either completely retrain the model with data up to the current time point or incrementally update the model parameters with the new data. 

There is no reason to apply reinforcement learning unless it is necessary.

If every data point does not have a ground-truth label, then reinforcement learning might be appropriate. Then you can not use Random Forest or any other supervised learning technique. You'll need to pick a Reinforcement Learning technique that can handle sparsely labeled data.