到处 is the adverbial which means "everywhere" the subject(people人们） is omitted

最近，到处在展开自发性的募捐活动。 Recently, there were spontaneous fund raising events everywhere. Obviously，到处 is a adv.

You can write in this way: 最近，自发性的募捐活动在到处展开。 In this way, you will recognize that the subject is 募捐活动. That's the logic of the sentence.

This kind of sentence is called 存现句 in Chinese.In this kind of sentence, the subject is usually omitted,so pay attention to the adv. and don't mistake it for the subject.

到处 is adverbial. The subject is omitted.but it can be 'country everywhere' or 'province everywhere' or 'city everywhere' or 'people everywhere' or sth else. what it is depends on the situation, but the subject in this sentense is not important,because the sentense's or the speaker's point is emphasizing the activity is wide.