The Karaite website "Hannukah: Have the Jewish People Been Celebrating a Pagan Holiday?" gives sources about the question. However, it does not give a definitive answer. The Karaite calendar shows no holiday during the period that Chanukah would fall in 2014 as well as 2015. It does show Rosh Chodesh Teves (as 10th New Moon). This would seem to imply that they do not celebrate Chanukah.
I checked both 2014 and 2015 to see Pesach and found that neither Purim nor Chanukah are shown on the calendar
UPDATE I did find references to Karaite celebration of Purim, with the dates listed at a different Karaite site, but there is no reference to Chanukah at all. The calendar includes the fasts of Tzom Gedaliah, Asara B'Teves, and several others in Tamuz and Av that are different from the ones that we have. I do not know if this is the same or different than the other calendar that I found above. This calendar includes the modern celebrations of Yom Hashoah and Yom Ha'atzmaut as well.
It appears from the references that Karaism accepts all celebrations that existed before the complete Tanach was canonized by the Anshei Knesset Hag'dolah. This would mean that Purim (Megilas Esther) is included but Chanukah is not (since it occured after the Anshei Knesset Hag'dolah no longer existed).
I do not know if these web sites are that of the real Karaites or the sites of modern creations.