My convection/microwave quit rotating.

This appears to be an Australian model and these are the only spares I can find. They do sell the latches. If the hinges are damaged and causing this, you will feel movement. Looks like it is time for a new one.

https://spares.bigwarehouse.com.au/index.php?cPath=46427_53341_57047

https://www.google.com/search?q=sharp+R-9X56+replacement+hinges

..

I really doubt that you can even get parts.

I was unable to locate a service repair manual specific to your microwave; sorry.

You'll must likely have to find a Sharp service and repair center.
I've attached a link that contains the user's manual to your microwave...it has the phone number and address of Sharp; perhaps you can phone them and ask where a repair center is located.

This is the diagnostic check list Sharp recommends before calling for service:
SERVICE CALL CHECK
Please check the following before calling for service:
1. Does the display light up?
2. When the door is opened, does the oven lamp come on?
3. Place one cup of water (approx. 250 ml) in a glass measure in the oven and close the door securely. Oven
lamp should go off if door is closed properly. Press the INSTANT COOK/START pad once.
A. Does the oven lamp come on?
B. Does the cooling fan work?
(Put your hand over the rear ventilation openings.)
C. Does the turntable rotate?
(The turntable can rotate clockwise or counterclockwise. This is quite normal.)
D. After one minute, did an audible signal sound and COOK
indicator go off?
E. Is the water inside the oven hot?
If "NO" is the answer to any of the above questions, please check your wall socket and the fuse in your meter box.
If both the wall socket and the fuse are functioning properly, CONTACT YOUR NEAREST SERVICE CENTRE
APPROVED BY SHARP.

Sharp Carousel 395Y Operation Manual And Cookbook

check the door lever

you might need to replace the door switch

Is it totally dead? No display? Or just does not cook? If it made an unusual noise before it 'died' something probably shorted out.... You need an ohm meter to troubleshoot. If the fuse is open, switches and HV circuit need to be checked before replacing fuse. Sometimes on the convection Microwaves (not just Sharp) the fuse will 'rust' and open rather than blow. Or the fuse holder can over heat and fail.

My sharp microwave convection oven (R-1870) runs for 59 seconds and stops. If I push start it runs for another 59 seconds and stops. How can this be fixedd?

The damper should closed when in convection mode and open when in the microwave mode, to keep the heated air inside the cavity.

at highvoltage section wiring may bae problem, check wiring and connectrors of magnetron and capacitor and transformer.

Check that the plastic connector from your motor to the turn table is not broken , mine broke and it took weeks to solve the problem. Try rotating the part that protrude into the oven. If you can turn it by hand it is broken, lucky cheap to replace