Here is my image of three entities, with ORDER and PRODUCT having many-to-many relationships:
I understand that I have to remove transitive dependencies to turn it from 2NF to 3NF. Does this mean I need to create an associative table between ORDER and PRODUCT with OrderID and ProductID as composite keys, and OrderQuantity as an attribute?
Do I have to do the same thing with CustomerPONumber (which is an attribute created by the customer themselves to keep track of their order and kept on their system)?
Your tables are in 3NF as far as I can see, only that there is no relation between orders and products, yet :-)
Yes, add a table to relate order and product just as you describe.
The CustomerPONumber seems to be in the correct place already. (It's rather easy to see this: you won't store the same number twice, so there is no redundancy, which is what normalization is about mostly.)