I will explain how AUTOBOX ( a piece of multivariate time series software that I have helped to develop) implemented a procedure to accomplish this and perhaps that will help you. Given a model which partitions observation to signal and noise ( hopefully white but not necessarily so ! ) for this discussion. Make a 1 period prediction and then bootstrap the residuals and incorporate any auto-projective structure to obtain a probability distribution. Now compute a second period out probability distribution. Note that no assumption about normality is required here .. the probability distributions are what they are.
One can then sum the elements of the two probability distributions to obtain a probability distribution of the sum.
If one has causal series that need to be predicted in order to predict the outcome series this can easily be included by incorporating the elements of each forecast for each series into this composite. In this way more realistic limits can be found for the dependent series which explicitly incorporate the uncertainty in the user specified predictor/causal series and not naively assumimg particular values for them.
I don't use Random Forest but prefer a more classical disciplined model building approach leading to an ARMAX model incorporating identifiable memory structure and latent deterministic structure waiting to be identified .
If Random Forest is not amenable to the aforementioned strategy i.e. re-sampling et al then you might want to look elsewhere for a viable solution.
I reached out to the web to find info on RF https://www.google.com/search?source=hp&ei=-_yMXLWXH66xggesxL7IBw&q=random+forest+explained&oq=random+forrest&gs_l=psy-ab.1.8.0i10l10.1903.6290..9547...0.0..0.91.1010.15......0....1..gws-wiz.....0..35i39j0i67j0i131j0j0i131i67j0i20i263.4OXyre3bYoM and got ...
"Strengths and weaknesses. Random forest runtimes are quite fast, and they are able to deal with unbalanced and missing data. Random Forest weaknesses are that when used for regression they cannot predict beyond the range in the training data, and that they may over-fit data sets that are particularly noisy."
so it would appear that no forecasting is possible using RF .