This is a very frequent question here.
Yes, you can buy the half fare tickets without being in possession of an actual Half Fare Card.
The Swiss Half Fare card must be latest valid the day you are going to use the public transport for the first time, this can be the first day of your stay in Switzerland. There is no need to activate it earlier.
During ticket inspection you simply show the ticket together with the Swiss Half Fare Card (SHFC).
When you buy tickets or the SHFC on sbb.ch as a logged-in user (not as guest), and you later use the SBB Mobile app with the same account, all tickets will be display in the app, including the SHFC.