Not usually. Most decent VFDs have electronic short circuit and ground fault protection for the LOAD SIDE, so when something happens there, that acts faster than the line side fuses would.
What most likely happened is that on small drives, there is a little current limiting resistor in the DC bus capacitor circuit that is used to limit the charging inrush current of the capacitors when first energized. That "pre-charge resistor" is then bypassed a second later once the capacitors are charged up. If the bypass contact welds closed, then the pre-charge resistor is no longer functioning and the inrush of the capacitors causes the fuses to clear. One thing that often causes that is if the VFD is operated by someone opening a contactor/starter ahead of it every time they run the machine. VFDs are generally not made for that and the manual will warn you not to do it, this is why.
As to repair or replace, it will be best to just replace it. On most small 5HP VFD, finding and replacing that pre-charge bypass relay is no simple task, it is often now just a surface-mount component on the single electronic board that is the VFD. Replacing a comonent on a surface mount wave soldered board is not for the novice electronics repair person, and that's assuming the VFD mfr will share the PCB drawings with you to be able to identify it.