If I study as an exchange student in Mexico or Spain, can I apply for a Taiwan Entry Permit directly there?

The first way is: you have to apply for the Taiwan Entry Permit before leaving the country, just print it and take it with you. The validity period of the Taiwan Entry Permit is 90 days. You should calculate the time and after the validity period of the Taiwan Entry Permit, You can also stay in Taiwan for 15 days if you enter Taiwan on the last day; another way is to consult the Taiwan Economic and Cultural Exchange Center in Mexico or Spain to see if you can apply for a Taiwan Entry Permit. Generally, it can be done in foreign cities. It depends on which country you go to whether you can apply for it. If possible, you can apply for one from abroad, print it and bring it with you, and then you can enter Taiwan directly without returning to your country. With a Taiwan Entry Permit, you can enter Taiwan directly from abroad.

Hope this helps