Are international financial institutions still necessary today?

This is a very broad question, and since the exact vertical within the financial system was not asked, will answer broadly.

International financial institutions usually enable trade, commerce, and banking across various geographies. This is done predominantly by having book balances with other banks (correspondent banking) as well as access to local markets thus being able to comply with local laws and provide access to markets.

As an analogy, they can be considered as financial routers that connect disparate financial systems/geographies together.