There are only a few countries with which
Taiwan maintains diplomatic relations because
of mainland China’s objections to
anything other than the 'one nation' policy.
Thus, there are no real 'embassies' in Taiwan.
These unofficial organizations can issue
visas, replace lost passports and handle
other issues for citizens of their respective
countries.

However, since they are not real
embassies, they often have to send
visas/papers overseas for processing which
may take weeks. It is advisable to check
with your particular county's organization
or 'embassy' for the current political
situation before leaving your passport with
them if you are in a rush or have a visa due to expire shortly.