Well for starters there are diseases that get nick named 'Vampire Disease'. Porphyria being the most well known. The actual disease of allergy to Ultraviolet Radiation is called Solar Urticaria, and is pretty rare. I have this disease myself and while it is horrible, it's not life threatening.
Most of the 'historical' reasons for why Vampires burst into flames is mostly Christian/ Hollywood magic. These were vile 'demonic creatures where anything light and holy would destroy them. Crosses, the sun, holy water, etc.
In a more modern setting the moon could definitely effect vampires. You could write something that during the Full Moon your vampires are weakened slightly, but they aren't incapacitated.
All in all I think it really depends on the context of your story. Is this a modern story where science can debunk things? Or is it a fantasy story where any reason could cause it? You could have it where for some reason the UVA/UVB light is damaging on a cellular lever for some obscure scientific reason. Or it could be that the Vampires were children of the moon and humans children of the sun, and the sun is a vampires weakness and the moon is the humans weakness and maybe they avoid going outside in the moon because of it. There are all sorts of possibilities! :D
As far as a scientifically speaking you might want to go the route of a severe allergy, but that the don't exactly burst into flames(maybe the bursting into flames came about as a fishermans tale so to speak, where each time it was retold the story was exaggerated). Maybe their bodies are extremely sensitive because they have a heightened immune system(immortality explanation as well ;) ) when their body absorbs the Ultraviolet radiation it interacts with the mast cells in the skin. The mast cells release histamine and can cause a severe anaphylactic shock!
My best suggestion is to take a look at these already existing diseases: Solar Urticaria, Mastocystosis, Mast-cell activation disorder, Xeroderma Pigmentosum, and of course how or why the body might produce an excessive amount of histamine! :) I hope that helps!!!
Edited to add: Conditions such as Solar Urticaria for example have a wide range of sensitivities. For instance, my sensitivity is on the severe end. I break out in hives from being under light bulbs that emit UVA. All of the bulbs are LED in my house. All of my TV's and computer screens are LED backlit. Lights that emit UV include fluorescent, incandescent, halogen bulbs and more. So depending on the sensitivity your characters may have to wear a special suit. I myself have to wear a special suit to go out in the sun. Even going out at night under a full moon is dangerous for me. I've had reactions under the full moon at night before. But I do always carry my epinephrine with me just in case!
You can also look up the Xeroderma Pigmentosum suits that they have to wear for some inspiration or other ideas. Those are extremely bulky suits but the suit that I wear is composed entirely of a double layer of UPF 50+ rating material(check out the link for what the ratings mean). Although in a fantasy story it would be completely plausible that they have a type of cloth that is such a tight weave it blocks out 100% UV so they don't have to wear a double layer!