What does Paul mean when he says he is "filling up what is lacking in Christ's afflictions"?

I think it's plain from the context and the wider teachings of Paul that he does not mean that he fills up by his sufferings what is lacking in the atoning worth of the death of Christ. When Jesus says, "It is finished" on the cross, all that had to be done to pay for the sins of all God's people was done. And nothing could be added.

So when Paul says, "By my sufferings I fill up what is lacking in the afflictions of Christ," what he means is that the one thing lacking in the sufferings of Christ is the offer of those sufferings in person to those for whom he died. So in essence he is saying, "Alright, I will take the message of the cross—where Christ suffered for lost people—and in my own missionary sacrifices I will take that message to them and say, 'In me behold the love of God, as I sacrifice to come to you and preach to you, and risk my life and imprisonment to preach the gospel.'" That's the thing that he provides that is lacking.

Christ cannot personally offer himself to people today. In and through God's people—especially missionaries—he offers himself to them. And so they fill up what is lacking, namely, the personal presentation of the sufferings of Christ in their own bodies.

What does that look like for those who may not be called to missions?

I don't think that text can be stretched to mean that everybody should suffer in the taking of the gospel to an unreached people group. No. I think it can be stretched to mean—though I don't think it is really a stretch—to say that we ought to love people in a way that takes risks. And by our risking they can see, in our willingness to be inconvenienced, lose comfort, or even suffer for them, the very nature of the afflictions of Christ that were offered for them.

So this text does have an application to everybody. The kind of love in which people will see the atoning work of Christ is the lifestyle of believers who are willing to take risks and make sacrifices in order to bless and do good to others.