As AdamO suggested, use the median from the original dataset.
The intuition is this: the data you used to develop the model should be large enough and representative to production distribution. Otherwise (production data is dramatically different) the whole model building process will not make any sense. If the model building data is representative and large, why bother to add few new production data?
Let's have a concrete example. Suppose we are building a model to predict housing price and the feature is number of bed rooms. The model building data (we can divide it into training and testing to build the model), has the median of 3 bed room. This numbers should be representative to the housing market overall. (say we had 500K data points /houses to build our model)
Now assume in production time, the first day is a special day and we have luxury 10 houses feed into the model, and these houses has ~8 bed rooms. In second day, that there are 5 houses with missing value.
Intuitively what would we do? To me, it is intuitive / natural to use 3 for imputation, because this number is calculated from a large representative historical data.