Mom is 84-years-old and bedridden. Breathing is shallow, and there is phlegm in her throat. She is unable to cough. What can be done remove the phlegm?

If you have Hospice they should be able to help with medication and other methods. If you do not have Hospice it is time to call them in.

You can roll her on her side and the secretions should drain.
You can take a swab and remove any secretions that are in the mouth. Do NOT put the swab in the throat as that can cause her to gag and possibly vomit and that may lead to Aspiration Pneumonia.

I sure hope your mom is already under hospice care, and if not call them today.
They will provide a suction machine like already said and can prescribe Scopolamine patches to put behind her ears to help dry up the phlegm.

If she is not on hospice, you should have her assessed. It is covered by Medicare.

If she is on hospice. They can bring out a machine that can suck out the secretions so she doesn't have to swallow them. They also can prescribe her medicine that will dry out thw secretions so there is less. My dad has a lot of excess fluid from cancer and heart failure and he uses a machine and takes medicine before bed now. He used not be able to lay down and would cough a lot. If she is not on hospice maybe there is still a way you can get access to one of those machines through a medical equipment warehouse or something. Or her doctor can maybe prescribe one.

There are devices to suction the back of the throat, which can give temporary relief. There are also medications to dry up secretions but I caution you they can have very unpleasant side effects as they dry up everything everywhere and I personally wouldn't recommend them. Positioning her on her side or helping her to sit in an upright position can help.
When my mom reached this point she was near the end of life and the phlegm was due to her inability to swallow her own spit any more, which eventually lead to aspiration pneumonia. Has she been seen by a doctor? Is she on hospice?