I will be going on a trip to Taiwan and will be going around multiple cities. There is a high speed rail system that I would like to purchase tickets for in advance. My trip is still 3 months away. Can I do that right now online?
No but you will be able to later. The official site is here and they sell tickets up to one month in advance.
Should you want to have your tickets and expect no issues with connections, you can buy them when your trip is a month away and you will have your reservations and train schedule in hand.
Otherwise, there is no problem buying them the day of. They even tell you when space is limited and allow you to book for the next one which is usually 20-40 mins later. If you buy in advance, then you will not be able to know if that train will be crowded ahead of time.
Just last month, when buying a ticket they told me that the train had only middle seats left and asked me if I wanted to book the one 22 mins later, which I accepted and was more comfortable for it. Given the High-Speed Train stations except for Taipei are far out, it is possible to miss your scheduled time should your transport to the train station be delayed.