Absolutely yes, Vanilla RNNs can be very good predictors for time series data. You can check this detailed official TensorFlow on time series forecasting, for example.
As they said in part 1:
the model will be given the last 20 recorded temperature observations, and needs to learn to predict the temperature at the next time step
This also gives you information about how to structure your data for training. Given an input sequence of given length, predict the next
n steps of the series - where
n can be 1 or more. The choice is very task-specific.
Assuming you are working with univariate time series (i.e. single series with no other variables) then you can shape your X_train as:
( number of observations , length of input series , 1 )
If you are working with multivariate datasets, then change 1 with your number of explanatory variables.
At each training iteration, the model will look at input data, and the actual outcome, and try to "learn" a mapping between x and y.