Do you feed him when he wakes up in the night? If you do, does he take a full feed and act hungry or does he just suck for a few minutes to get back to sleep? If he's taking a full feed, he's probably hungry. If not, then something else is bothering him. How big are his daycare bottles? At 5 months, my DD was nursing 5 times a day, so it seems like it cound be enough.
Early awakenings are not totally unheard of at this age - and for my DD, early awakenings at this point turned out to be due to teething. If it's a hunger issue, you'll have to figure out how to get more calories in him during the day - I can certainly understand wanting to hold off on solids if you have allergies in the family - maybe cluster feeding would help?
Good luck!