Can I get my prominent ears pinned back without having surgery?

Not possible. Ear surgery, or otoplasty, is usually done to move prominent ears closer to the head or reduce the size of large ears.
Otoplasty is best. One device is an ear band with a molded earpiece, but this is really on for babies, when their ear cartilage is still moldable. The device has to be worn all day. An otoplasty can be useful to correct issues like a prominent ear where the entire ear is pushed forward or a cup ear deformity where there is a lack of one of the normal ear folds. Consult a facial plastic surgeon for a consultation.