VFD for a 3-phase delta-connected 415V~ motor
When a motor starter is to be replaced by a 3-phase 415V~ VFD, can the existing motor be used without changing its winding configuration (star/delta)?

Yes. VFDs have the same nominal output voltage as input voltage. If you have a 415V supply, you buy a 415V input - 415V output VFD. You connect the VFD for 414V operation. It makes no difference whether the motor was originally connected star or delta, it must be connected for 415V operation. When you disconnect and reconnect the motor, you must get that correct. Just as if you were to change from a star-delta starter to a direct-on-line starter.
The only exception is the connection of the motor for operation above the original frequency, for example 87 Hz rather than 50 Hz. However that is a different and more complex question.