China was pressuring the manufacturers, i.e. "Taiwan is part of China, so you should sell to us and we will give them to Taiwan." Then they tried to pressure Taiwan for concessions to get the vaccines to save their lives. Taiwan ordered vaccines, only to have them not show up. They had to trade chips from TSMC for access to vaccines.
This is the same as China using political influence and corruption to keep Taiwan out of the WHO.
The Taiwanese government knew exactly who to call up to order the BioNTech vaccine - Fosun. For political reasons, they refused to do so.
Instead, they knowingly tried to get BioNTech to breach its contact with Fosun. When BioNTech predictably decided not to, the Taiwanese government acted outraged and made a big media affair out of it.
> They had to trade chips from TSMC for access to vaccines.
No, that's not what happened. The founder of TSMC stepped in to buy vaccines from Fosun, so that the Taiwanese government could claim it hadn't done business with Fosun, a mainland company. This was a face-saving measure for the Taiwanese government. After declaring that they would never buy vaccines through a mainland company, they couldn't easily climb down, so the TSMC founder acted as an intermediary.