Is it permissible to work online through websites that stipulate acceptance of conditions that violate Sharia, such as imposing arbitration by non-Sharia courts in case of dispute, and is this considered a matter of necessity, even though I am financially well-off?

There is no objection to accessing websites that include conditions contrary to Sharia (definition) — such as resorting to non-Sharia courts — due to the widespread affliction (balwa) of such conditions and the difficulty in avoiding them. The Muslim intends to affirm what conforms to Sharia, not what violates it. Furthermore, these conditions are often merely perfunctory. The fact that a person is well-off does not prevent them from dealing with these websites if they have a need to do so.