Hello Azo
From my experience, depending on the payment provider and card used, it may take up to 10 days.
There is no need to purchase the Swiss Half Fare Card that far in advance. You could also purchase it 5 minutes before travel. Important is that the card is valid the moment you enter a public transport vehicle, not before.
You are not required to be in the possession of the (Swiss) Half Fare Card to purchase half fare tickets. The ticketing system will not actively check if you hold a (Swiss) Half Fare Card.
Kind regards,
Roland