Hey!
Congratulations on the VS offer!!
3-5 coffee chats a day does sound like quite a lot - on some of my schemes, I didn't even have that many across the whole week! I would personally treat it as a rough guide rather than a strict expectation, as you will have work to do and that should always come first. If you are busy, focus on doing that well and just fit in coffee chats where you can. Of course, if the firm is encouraging that level of networking, it's worth making an effort to arrange a good number, but I would still prioritise the quality if your work over hitting a specific number of chats.
In terms of approaching people, you definitely don't need to just walk into offices unannounced (although this can be a quick way to arrange one if they are in your team!). I personally found it easiest to send a brief email which said something along the lines of: "Hello X, My name is Y and I am on the vacation scheme this week, currently sitting in Z department. I would love to hear about your work if you have some time this week". Most people are very open to it, including associates and partners - they know that you'll be at the firm, and will want to learn. I'd also make sure to put it in their calendar once you have arranged it, so that they don't forget!
I also stressed about what to talk about in the coffee chats, but I would try not to overthink it! You don't need to lead with a commercial news story unless it comes up naturally. It is completely fine (and usually expected) to ask questions about:
- What they actually do day-to-day
- The types of deals or matters they work on
- How they got into their practice area or joined the firm
- What they enjoy or find challenging
If you can link in a bit of commercial awareness, that is great, but it isn't a test so I'd try not to force it. It is more about showing a genuine interest in the individual, and being easy to talk to.
The overall aim of coffee chats is to learn more about the firm and to come across as engaged and curious. You don't need to be overly strategic about it, and a few good conversations a day alongside your work should be more than enough