absolutely - we did all our shopping on this and regularly picked up multiple 25kg bags of coal and kindling on it for over a year, and before that it was on circuit as a courier bike. you can secure a box on, but i always preferred the versatility of using an ikea bag as they're more flexible in terms of load (which is why we have an omnium and not a bullitt). you just strap it on using the inner tubes.
and going to the launderette (more than enough space for two ikea bags):
i'll even throw in the mudguard. we move in two weeks, someone please take this off my hands!