They do have a phone number, source: https://www.swtor.com/info/faq
USA: 1-(855)-345-2186 (Support available Monday - Saturday from 09.00 CST - 18.00 CST)
UK: +44 (0)208 1969 890 (Support available Monday - Saturday from 15.00 UTC - 00.00 UTC)
France: +33 (0) 4 81 68 10 51 (Support available Monday - Saturday from 10.00 UTC - 19.00 UTC)
Germany: + 49 (0)221 8282 9212 (Support available Monday - Saturday from 10.00 UTC - 19.00 UTC)