So I went to the ER last night and found out I have a cyst on my right ovary and I'm in severe pain and really sick, can I have it removed?

Cyst ovary. Ovarian cysts can be removed. You need to see the ob/gyn and discuss this at that visit.
Ovarian cyst. Ovarian cysts frequently resolve on their own without surgery. The pain and sickness is frequently from small amounts of bleeding which irritate the pelvic or abdominal wall. Discuss treatment options with your gynecologist.