Yes, your insurer must accept your January payment to start your coverage for next year. Insurers are no longer allowed to require people who owe back-due premiums from the past 12 months to repay the premium debt before they will renew coverage for another year. Whether you actively renew or auto-renew, the process will work the same way.
Keep in mind that insurers are allowed to cancel your coverage if you don’t pay premiums. In most cases, if you fall behind, the insurance company is required to offer you a 3-month grace period to catch up on back-due premiums. The insurer must continue to pay your claims during the first month of that grace period. If you haven’t caught up on premium payments by the end of the grace period, the insurer can cancel your coverage retroactive to the first day of the second month of the grace period and deny claims that you submitted during months 2 and 3 of the grace period.