Proficiencies require training time and money. So it can happen anytime they gain the money to do so. However, due to other expenses, they often have to decide as a group when and who will get to train. This means they might have to sacrifice buying that new plate armor or living in a cheaper room. plus its usually a great risk to adventure long enough to gain the required coin to afford to train.
As a DM, you can give this as little or as much extra flavor as you would like.