The longest Israeli school vacation is the summer break from the 1st July to 26th August, this is the worst time to visit Eilat.If you have to go during that time them make it is the first or last weeks.
Other school vacations in Israel vary slightly each year as the Jewish calender (which determines the school holidays) goes according tot he moon and not the sun, like the Gregorian calender. So September is a holiday month with the Jewish New Year, Yom Kippur and the week long Sukkot. December has a week long Hannuka holiday. In April or May there is a two week long Passach holiday (often at the same time as Easter). Check out the exact dates before you go but basically these are the months will be the most crowded and the hotels will probably be at capacity. There are other shorter holidays which shouldn’t affect your vacation in Eilat.