Difficulty understanding sentence in The Economist
From a strictly grammatical perspective, you could interpret "in hospitals and care homes" as modifying the phrase "who have had their jabs" and as indicating where (geographically speaking) the "those" who have received their their shots were inoculated. But I think that the author's actual intention was to link "in hospitals and care homes" directly to "those."
Logically, the point that the writer means to emphasize here seems to be the continuing vulnerability of certain people—namely those in hospitals and care homes—despite their having already received vaccines, not the continuing vulnerability of people who happened to receive their injections in a particular setting.
A clearer way to express this idea would be to revise the text as follows:
That leaves two reasons for passports at home. One is to enforce vaccination in places where an infected person could harm people who remain at risk despite their having already had their jabs—in hospitals and care homes, for example. Similar thinking has led some countries to require proof that any person working with vulnerable individuals has no criminal record. ...
This revision improves the clarity of the writing by replacing competing plural entities ("infected people" and "those who have had their jabs") with a singular endangerer ("an infected person") and the plural endangered ("people who remain at risk"). But how to say something more clearly is ultimately a copyediting issue—not, strictly speaking, a question of grammatical correctness.