The short answer to this question is "the off season," but in all likelihood, the time of year is the fall. In the first chapter of The Wide Window, the taxi driver that takes the Baudelaires to Aunt Josephine's house explains that Lake Lachrymose is a resort town and looks abandoned because it is the "off season"; it will become very crowded when the weather becomes better. From this, it is possible to infer that the time of year is the fall. In chapter 2, the Baudelaires are looking forward to a meal of hot soup because it is "a cold night," another clue about the time of year.
Perhaps the best argument for the story taking place in the fall has to do with the approach of Hurricane Herman. Hurricane season in the Atlantic takes place in the fall, and even though Lake Lachrymose is only a lake, as the taxi driver explains, anything can happen with such a large body of water. In chapter 4, when Count Olaf breaks the big window in the library, the cold air rushes through the window into the room, "rattling the bookshelves and making the children shiver." Reference is made to "gloomy" mornings, suggesting both the approaching storm and the shortness of the daylight hours.