"Well, we'll mange," said Mrs Weasley, but she looked worried. "I expect we'll be able to pick up a lot of Ginny's things second-hand."
I think "pick up" here means "to buy". However, Ginny's things second-hand looks odd. Maybe, second-hand, as an adjective, is used as a complement? Otherwise, I think it might be better to put it as: pick up a lot of things second-hand for Ginny. Does it make any sense?
It's an adverb here:
adverb
On the basis that something has had a previous owner.
‘tips on the pitfalls to avoid when buying second-hand’
Oxford Dictionaries
It sounds perfectly fine to me. So does your proposed rewrite, which also uses it as an adverb. It means that Mrs. Weasley will be either shopping at the wizarding Goodwill or perhaps see if she can get them from her family.
Ginny's things means "things Ginny needs" or "things for Ginny".
To pick up is an informal way of saying "purchase" or "buy".
second-hand characterizes the shopping for or the purchase of a pre-owned item.
When shopping for furniture for their rooms, university students typically shop second-hand.
The university student bought a desk and chair second-hand.
The opposite is "new":
When shopping for furniture for their rooms, university students typically do not shop new.
Few university students buy their furniture new.
Both can be used adjectivally:
The university student bought a second-hand desk and chair.
The university student wanted to buy a new desk and chair but settled for second-hand.
I might be misreading your question here but the gist of the sentence is: Mrs Weasley is worried about money but she thinks they will be OK because she thinks they will be able to buy everything they need used or second-hand.
