I've been wondering if my plant will be able to produce peppers in this container?
Yes, a plant can produce many chilli pepper in such containers: it is common to use small pots (especially for shops).
But for me, the plant is maybe too much watered and fertilized, on the other hand, it is still a seedling, so it could be ok.
An other worry from me is about your location (windows with condensation?): if it is fall and cold (and not much sun), you may have few flowers (and so peppers) until springs.