The claim is plausible. Let's break it down into several components:
Q: Are monkeys capable of stealing wallets or learning to use vending machines?
A: Certainly. There is ample video evidence of monkeys stealing wallets and using vending machines. Token-based food vending is common practice in studies of animal behavior.
Q: Are simians capable of learning how to use vending machines on their own?
A: Yes. See "Socially biased learning in monkeys" by Fragaszy and Visalberghi in Animal Learning & Behavior (2004) 32: 24 among similar studies.
[...] we presented two different food-providing apparatuses to two groups of monkeys. One apparatus functioned like a vending machine [...] When a rod chained to the apparatus was inserted far enough into a transparent tube, a small quantity of sunflower seeds fell into a cup below the tube [...] Several monkeys in each group learned to solve these problem (p.25).
Q: Do monkeys in Mino near Osaka participate in the above activities?
A: There are anecdotal accounts of that happening on TripAdvisor and Travel Wire Asia.
Incidentally, chimps (not monkeys) can teach the use of vending machines to other chimps and save money and transfer tokens for the use in vending machines. Dogs have been observed to ride the subway to and from a specific location. Animals are pretty smart!