B737 First Officer reported trailing edge flaps failed to extend on approach. The flight crew performed a go around to trouble shoot; then use the alternate flap extension procedures. This resulted in a flap asymmetry condition. The flight crew received vector back to make a precautionary landing at destination airport.

Narrative

On approach into ZZZ the Captain called for flaps and after selecting the desired flaps setting; we noticed the flaps remained at 0. We then decided to go around to troubleshoot the problem. We notified ATC that we would need delay vectors to run the necessary checklists. We ran the QRH checklist for Trailing Edge Flap Disagree; the checklist had us setup to use the alternate flap extension procedure. After getting flaps 15 landing data and brake cooling data for Runway XXR we had ATC vector us back around for another approach. The leading flaps and slats extended to the full ext position using the alternate flap switch. When we went to extend the trailing edge flaps; only the left began to extended and I immediately stopped extending the flaps as we now had flap asymmetry. We notified ATC we could not fly the approach and would need a longer runway at ZZZ1 along with delay vectors. We began to run the QRH checklist for Trailing Edge Flaps Asymmetry as directed by the QRH 'Trailing Edge Flaps Disagree' checklist. As ATC began to vector us for ZZZ1 dispatch sent us an ACARS message that they ran the numbers for a Flaps up landing and that XXR at ZZZ would be suitable for a 0 flaps landing. We ran the numbers as well and agreed with dispatch. We then completed the QRH checklist and received Brake cooling data as well. We notified ATC that we could in fact accept XXR at ZZZ and received vectors for the visual approach. We Performed a 0 flaps landing on Runway XXR at ZZZ. As directed by the brake cooling data; we proceeded to Pad XX and waited the 70 minutes for the brakes to cool with fire trucks monitoring. After the 70 minutes we proceeded to Gate XX and deplaned. A maintenance discrepancy was written in the maintenance log.