According to the Oxford English Dictionary, the first phonetic use was by the poet Ernest Dowson, who wrote a letter on 21 February 1889 in response to an invitation to watch the Varsity football match, saying "I absolutely decline to see socca' matches but will matinize with you whither you list" (ie There's no way I'll go to the football but will meet you on Saturday morning somewhere else). This indicates the word was in verbal use at that time but not normally written down.
However, also according to the OED, it was written as 'socker' in October 1891 and for the first time as 'soccer' in January 1894. This seemed rather late, so I searched through online newspapers for earlier mentions. Although there are inherent difficulties and false leads using optical character recognition, I came across the above paragraph under the heading 'Notes from Oxford' in the North Wales Chronicle of 14 March 1891. The unnamed correspondent, who had a weekly column in the paper, gave a few examples of Oxford slang to tickle the reader's ingenuity: "What does he think of - ragger, togger, roller, footer, rugger, brecker, soccer, fresher &c?"
Could this be the first written use of soccer, three years earlier than recorded in the OED? It seems unlikely that a local paper published in Bangor would get there first, but the nature of a column like this was to entertain readers with little snippets. So here's a challenge - can anyone find an earlier use?
Update, from the Boy's Own Paper of 6 April 1889: "In Varsity patois Rugby is yclept 'Rugger', while Association has for its synonym 'Socker'." That seems to be the first published use of the word, but not yet with the later spelling.