It may be problem with metal clips holding the SD card. Try to adjust them. I think camera has internal memory to place a few pictures without SD card.

Try this guide to dissamble the camera: Fujifilm FinePix AX665 Lens Assembly and Gears Replacement Replacement But be careful and have rubber gloves on (it amy be shocking). I’m trying to fix same problem with my J30 old camera but all the black tape and grounding making it quite difficult to remove the lens safely.

Try installing FinePixViewer, maybe it needs this program to view pictures on PC