Your supervisor/trainee buddy will generally give you work. You’ll also have an tour of the department you’re sitting in on the first day – that’s when I’d usually mention I’d be happy to take on any work, so sometimes people will know they can come in and give you work. And sometimes I’d just offer it to people who came into the office.
Aside from that, I don’t think it’s necessary to get work from lots of people, at least in the sense of knocking on doors or searching for work (apart from Jones Day where it’s part of the training). There’s some qualifications to that, for example if you found yourself without any work – then I’d suggest seeking out work.
It’s a bit strange because you’re on a scheme without many highly-qualified students, so I know it often feels like you need to go above and beyond everyone else. But I’d think of standing out this way – just focus on doing well in each activity you do. That means work really hard on every task you’re given; be nice to every person you meet, from the IT staff to the secretaries to other students; and ask genuine questions during department talks. I think it’s better to get good reports from a smaller number of people you do interact with than spreading yourself thin or trying to get the attention of senior lawyers.