Coming in late here and waving hallo to everyone as a new poster. I've been reading the posts here with great enjoyment. :)
But, no, to the above. Church of England curates and vicars do not have to take a vow of poverty. For that matter, neither do Catholic Diocesan priests have to make vows of poverty. Those are only required for monastic orders, including the C of E Nonnatus nuns.
Mind you, your average parish priest whether C of E or Catholic might as well take a vow of poverty; the base pay these days for both denominations is pretty darn low, although they do get housing as part of their compensation. However, some have other sources of income and they don't have to forfeit that to become a vicar or parish priest .
I felt sorry for that young curate although I enjoyed Sister MJ's take-down of him. It made me laugh. As someone else said above, he needs to learn to deal with cantankerous old bats like her if he is going to grow up to be a vicar. Also, in a disadvantaged parish like Poplar, he would be living on a tiny stipend but would be entitled to accommodation provided be the parish. It is the vicar who is responsible for landing the poor chap on Nonnatus House. The sisters should have taken any issue up with him, not the very clumsy but well-meaning curate.
To clarify, historically there was huge inequity in what C of E priests (let alone curates who are at the bottom of the heap) were paid. If you think of Jane Austen novels, you have a few vicars, appointed by the diocese to a parish, a scattering of rectors, appointed by some local aristocrat to a "living" in their control, and a few put-upon curates. Some of them were wealthy, especially those rectors appointed by their rich relatives to prosperous parishes where they could collect tithes. Think of the appalling Mr. Collins in Pride and Prejudice. He was all set to inherit the Bennet property because it was entailed to male heirs. With the aristocratic families, the oldest son inherited the title, and the younger sons usually joined the army or entered the church and took over the living when it became vacant. Vicars in rich parishes, unlike Poplar, could also make a decent salary.
It wasn't until sometime in the very late 1960s or early 70s (I can't remember exactly when) all this was changed and salaries for clergy were standardized across the board by the C of E.