It means to snore (いびきをかく). Some verb in japanese only make sense when combined with its intended noun, so you can sometimes try searching for the entire phrase instead of each individual word.
This verb can be shown by clicking to "show collocation" under the entry for いびき, or search for "いびき" and just scroll down a bit xd. It also have its own entry as "いびきをかく" so as said above, try guessing the original form (change かいて to かく) and search with the entire phrase.
It's in the te-form.
いびきをかく＋て -- いびきをかいて. Note that this has become an adverbial phrase, and it is modifying the verb 寝ている. It explains how he or she is sleeping.
Te-form is a popular topic, so you can find the information on the internet. Regarding the combination (collocation) of いびき and かく, see Lyza's good explanation above.
Stumped again....
OK, I'm probably missing something obvious here--but I don't understand "かいて" in the following sentence.
ルームメイトがグーグーといびきをかいて寝ている。
Thanks for any help.