The correct answer is C. on.
Lavish means “spending or using money or resources in an extravagant or wasteful way.” So, if someone is lavish with their expenditure, they are spending money in a way that is extravagant or wasteful.
The other options do not make sense in the context of the sentence.
A. from: This would mean that she is spending money away from something. This does not make sense in the context of the sentence.
B. with: This would mean that she is spending money together with something. This does not make sense in the context of the sentence.
D. in: This would mean that she is spending money inside something. This does not make sense in the context of the sentence.