Have you tried to 'power reset' the camera at all?
Try the following and see if it works:
Remove the main battery, then turn on the camera and press and hold the shutter button for at least a minute, to try and fully discharge the internal backup battery. (it may take longer, depends on how much charge is in the battery)
The battery is an internal, non user removable, rechargeable battery that is used to maintain the camera's settings when you remove the main battery.
Once you are satisfied that you have held it for long enough, release the shutter button, switch off the camera and reinstall the main battery and then switch on the camera to see if it now works.
If it turns on you will find that the camera will be reset to a factory default condition and that the date and time settings etc will have to be reset.
When you reinsert the main battery allow at least 24 hours for the back up battery to be fully recharged so that the camera will retain the settings when you next remove the main battery.
Here is an image taken from the service manual for your camera.
(Click to enlarge for better viewing.)
Looking at the circuit (as shown in the image above) it seems as though the access LED is operated by the insertion of the SD Card,( as you already probably know).
My view of the circuit suggests that insertion of the card sends a signal to the image processor to perform checks on the card as well as turn on the access light. As the access light is always on as @oldturkey03 suggests the card reader may be faulty OR there is a fault further into the board.
Unfortunately for you the SD card reader is mounted on the 'B'side of the main image processor board. Also I cannot find online anywhere that supplies what I think is the correct replacement part. In the service manual the part number for your board is A-1789-527-A AM-022 BOARD, COMPLETE (SERVICE) (SLT-A33/A33L/A33Y but when I search online for this part to make it easier to replace rather than having to fix the card reader all that shows is this https://www.encompassparts.com/item/8356...) which actually looks the same but has a different part number. (Letter B at the end in lieu of A. It may be that the A & B sides are two boards but it could also be the same board being used on both sides and it is expensive if it is the wrong part.
Finally as I cannot find a disassembly guide for the camera online (as the service manual does not show it either) and to save you disassembling the camera without an adequate guide you may wish to try using a can of compressed air to see if you can 'blow' the card reader clear of any dust etc which may be affecting the card reader. Be careful how you use it. Do not apply for too long or too strong and do it with the battery removed from the camera.