Here's the story for those interested: http://time.com/4060037/titanics-last-lunch-menu-sells-at-auction/
There's lettuce and tomatoes and a few other things listed. This was obviously within a couple of days of leaving the US. But how would they have been able to supply vegetables over the rest of the journey later on?
You may be overestimating the time that the journey took -- the runs by Titanic and her sister ships were meant to take no more than five or six days in each direction, leaving weekly Wednesday from Southampton and Saturday from New York (the ships would rotate that schedule). Vegetables will keep perfectly well for a week or less, especially in cold storage.