Mount Elgon is climbable any time of the year, however you need to prepare for both rain and the cold.
The lush greenery is possible because of all the lovely rain that falls on the mountain.
Furthermore, agriculture in the area depends entirely on rainfall (there is almost no irrigation), so it is a blessing that there are two rainy seasons allowing for two growing seasons.
The months that usually see little rainfall are: June-August and December-March. However, showers are still possible during these months.
Mount Elgon rises over four thousand meters and so it is no surprise that things get very cold along the way.
Prepare adequately for the rain and the cold, and have a pleasurable climb!