Can "while you're at it" be used in non-physical contexts?

Yes, this is fine.

The point is that in your original example "While you're at it, could you get some milk?", "it" doesn't refer to the store, but the activity of buying a newspaper. (If "while you're at the store" were meant, the natural thing to say would be "while you're there".) "While you're at it", in this context, is a short way of saying "while you're engaged in that activity", so there is no reason it needs to be a physical activity.

I agree. It is OK to use for non-physical situations.

This app computes your total bill, and while it's at it imitates the sound of a 1950's cash register.

"At" doesn't have to imply physical proximity; it can also have other prepositional uses. It can be replaced with "about" in the phrase, "while we're about it." See definition 3 here, "used as a function word to indicate that with which one is occupied or employed."

I'm a software engineer and I do a lot of bug fixes. While I'm at it, I often try to do code cleanup, such as deleting code that isn't used, making logic simpler, adding comments to confusing parts I have already spent time trying to understand, and do several other things to make the code more readable and less confusing.

Oftentimes browsing questions and answers here on Stack Exchange, I'll answer a question, since I'm already here.

The phrase "since I'm/you're already here/there" has a similar connotation as "while I'm/you're at it". It implies a location, but it can also mean an action. In the 2nd paragraph above, the phrase really means that I've already read the question so I can answer it, rather than just that I'm already at the page that asks a question.

This can go further, too. "Since you're going to the grocery store, can you also go to the hardware store and pick up some more screws?" The hardware store might be next door, but maybe not. This is a little more abstract, but it's still "while you're at it". The action here is more generically "running errands", rather than specifically getting groceries.

These examples can also be used non-physically.

• While you're already deploying the mobile app updates, can you also deploy the new website updates, too?
• Since you're already in that app, can you fix these other bugs, too?
• You've been reading Charles Dickens? While you're at it try reading some Daniel Defoe.
• You're thinking about how to fix climate change? While you're at it, how about figuring out how to fix world hunger?

You can even have it mean physical and non-physical things at the same time.

• You're doing yardwork? While you're at it, can you think about what living room set you like?

Most of the time, these "while you're at it" tasks are related, but they don't have to be. My last two examples are specific to that idea. Climate change and world hunger can be loosely related, since climate change affects the ability to feed people, but simply fixing that won't also fix the socio-economic and political reasons people aren't getting the food and nutrition they need.

And the last one is an example of how you can do two things pretty much simultaneously without interfering with the other task. Yes, you need to focus on mowing the lawn so you don't also mow the garden, but you can still think about how the furniture is going to fit in the room, how the colors match the woodwork/carpet/paint/tile/etc., how comfortable they were in the store, how expensive it is, and all those other things without really taking time away from picking up sticks/trimming bushes/raking leaves/whatever.

We do these kinds of things without even thinking about it most of the time. While we're driving, we think about how our day went, what we're going to say at the meeting/interview/, how we want to get to 2nd base with our date, and more.

We even do it accidentally. You could be thinking about a problem at work and while doing so, you realized you can fix a problem at home using the same kind of solution, or you were filling out some paperwork and suddenly you realized just how much repetition it is and figured out exactly how a single form could replace the 5 current ones.

These kinds of things are all related to "while you're at it" and are definitely the kinds of non-physical contexts you ask about. It's not always asking someone else to do something or even that it's a conscious act, but something we do so naturally that we usually don't even recognize when we do it.

And to address the definition of "at", it's not that it only represents a place, but it also represents time, and many other things.

• I'm not at a good place in my life right now.
• I'll meet you at noon.
• He grabbed at the baseball, but couldn't catch it.
• You can reach me at my work phone.
• I've been thinking for a while and finally arrived at a decision.
• Why is your cat staring at nothing?

This is part of the reason why English is so hard to understand. Words can have so many different meanings that the context of when and how the word is used is almost more important than the word itself.