B737 flight crew reported being unable to retract the landing gear after takeoff. Flight crew ran checklist and resolved problem.

Narrative

As we climbed away from the runway at takeoff; and achieved a positive rate of climb; I retracted the landing gear. We soon realized; by landing gear light indications; and the ambient outside noise; that the landing gear did not retract. I notified ATC that we had a problem with the landing gear. I then requested priority handling; and told ATC we needed to climb to a safe altitude; and get delay vectors; while we ran the appropriate non normal checklists from the QRH. I chose to request priority handling to get the attention of ATC of our situation. Also; if we could not resolve the problem back to normal; we would have to return to ZZZ Airport to land; regardless of the outcome from then on. Once at a safe altitude; First Officer transferred the controls to me; and I continued to operate communications with ATC. This was because the landing gear handle and emergency gear extension door were at a closer proximity to him. First Officer ran the 'landing gear fails to retract' and 'gear disagree' checklists from the QRH. After he performed step 5 from the 'gear disagree' checklist (open and close the manual gear extension access door. Verify door is fully closed); the gear was able to be retracted; and all indications were now normal. At this point; I transferred the flight controls back to First Officer. I then told ATC that we were back to normal operations; and were no longer in need of priority handling. I also informed them that we could continue to our destination of ZZZ1 airport. (Note: all communication with ATC from the beginning of our non normal event; until our return to normal operations; was with ZZZ departure control). I informed dispatch via ACARS message of the situation; and sent an ACARS ELB (Electronic Log Book) report to Maintenance Control. I also called our flight attendants; and informed the purser of our situation; and informed her that we were continuing to ZZZ1 airport.