Yes, a learner driver can get insurance on their own car, as long as they have a valid UK provisional licence.
Learner car insurance allows you to practice in your own vehicle outside of lessons with an instructor, as long as you're supervised by someone who meets the DVSA requirements. It’s a good option if you plan to get plenty of private practice before your test. Some annual learner policies may also let you build up a no claims bonus.
Note: a learner driver must be supervised by a suitable person, aged over 25, who has held a full UK licence for at least 3 years, and will only be covered whilst being supervised to learn to drive with the supervisor in the passenger seat.