There are two reasons normally that the cassette door won't open when you slide the Eject button on the JVC GR-SXM320. One: the camera uses a powered eject feature. If the battery is low and the AC adapter isn't plugged in and powering the camera, this doesn't work. Two: if the LCD monitor screen is not open to more than a 60 degree angle, then there is an interlock that keeps the cassette door closed. This avoids the door hitting the screen. It's best to have the monitor at a 90 degree angle to the camera body when you open the cassette. Press Stop if there is a chance that the camera is recording to the current cassette.
JVC has the manual here: http://resources.jvc.com/Resources/00/00/97/LYT0465-001A.pdf
Note: since the camera is older, check that the AC adapter is getting warm when it's plugged in and connected to the camera. This checks if the adapter is working. Check if your battery pack is good. If you have the BN-V856U battery pack, you need the AA-V80U charger to charge the battery. The charger that came with the camera will not work.
Third party battery packs and chargers are still available for the GR-SXM320. Use the manual to look up the part number and Google for the battery or use the camera model number to find available parts. For example, a replacement BN-V11U battery is available at Amazon and here: https://www.bestbatt.com/JVC-BN-V11U-BN-V12U-BN-V14U-BNV-V18U-Battery-p/bbbnv11u.htm
I hope this helps.