Yes, you can use 'collect' in this way. We would be more likely to say 'pick up' or simply 'get' in this context, but 'collect' is fine. It is slightly more formal than the other two options - phrasal verbs and expressions with 'get' are generally informal.
I believe that this is a mainly British usage. Some weeks ago an American girl commented that she was 'weirded out' when she read a sentence about someone collecting their children from school.