Many pharmacy’s will sell you SSRI’s directly, some will refuse but generally if you dress respectively and have an old box or prescription you should get them. Obviously they’re not something to be played with or Google self diagnosed.
Most hospitals have GP’s or equivalent, usually under “Family Medicine”, no idea if they will prescribe or not. https://www.wanfang.gov.tw/p3_register_e3_english.aspx?deptcode=3700&depttype=B&deptdesc=Family%20Medicine&depttype2=T
Not sure why you’re having difficulty finding a psychiatrists, just looking at Wangfang’s page there doesn’t appear to be any difficulty making an appointment. https://www.wanfang.gov.tw/p3_register_e3_english.aspx?deptcode=3600&depttype=B&deptdesc=Psychiatry&depttype2=T
Tzu Chi in Xindian also has a large Psychiatric department and you can make an appointment online, https://app.tzuchi.com.tw/tchw/opdregEXD/OpdTimeShow.aspx?Depart=身心醫學科&HospLoc=4
Dr Wang appears to the doctor recommended in previous threads so might be worth giving him a try