Ideally, what department should handle credit applications for new customers?
Let me state that again--both departments work for the same company. So the end goal should be to do what is in the best interests of the company, not the department.
What would be best for the company is if there was a separate independent department that could perform the credit approval function. However, I realize that many companies, especially smaller ones, simply can't afford to pay for this department.
Keeping that in mind, you are correct in stating that a salesperson is the ambassador for the company but sales does have some responsibility in the credit approval process.
Specifically, the salesperson should perform an initial screen on potential customers by asking some questions and requesting credit information. In terms of analyzing the data and processing the credit approval request, it would be best for employees that are not in sales to handle this function.
If you don't have a credit approval department, than the billing or accounting department would make sense. It is very important to segregate duties in this situation because the sales department is likely less objective and more likely to receive incentives for closing sales whereas a separate billing or accounting department would likely be more objective and their work is not directly linked to closing sales.
In summary, the sales department should be responsible for the initial screening process but it is best to have a separate accounting or billing department handle the credit approval and processing functions.
Be sure to highlight to your colleague in accounting how important it is to have segregation of duties, especially with a high risk area like credit approval.