I am having water in my disposal back up into my dishwasher and side sink after I run the disposal. After running disposal, water disappears but after I shut off, it returns. The water backs up into dishwasher as well.

What can be used to unclog or fix this problem without calling a plumber.

You have a blockage or partial blockage in the sink drain line. The disposal is forcing water into the drain line and up the vent pipe. When you turn the disposal off the water in the vent pipe runs back down into the sink.

Try plunging. Put the drain stopper in the disposal side. Fill about two inches of water in the other sink and plunge rapidly. The pulling action of the plunder is more effective than the pushing action. If this does not clear the blockage remove the trap underneath the sink and see if the blockage is there. If not, snake the drain line. It sounds like the blockage if fairly close and you should be able to clear it with a manual snake.

If you are getting water back into the dishwasher you must not have an air gap and the drain from the dishwasher is not tied up high enough to prevent contaminated water from the disposal draining back into the dishwasher. Secure the drain from the dishwasher up to the bottom of the counter top.

I think you might just push the water back into your dishwasher by plunging. Cut to the chase and open the drain line at the trap. Check the trap and rod the lateral line going into the wall.

I am having water in my disposal back up into my dishwasher and side sink after I run the disposal. After running disposal, water disappears but after I shut off, it returns. The water backs up into dishwasher as well.
What can be used to unclog or fix this problem without calling a plumber.
Bob's correct. You have a blockage in the horizontal drain line in the wall. The table of the disposal acts like the impeller of a pump whe the disposal's full of water and you turn the switch on. If you run the disposal when the sink's backed up it will pump water back down and through the high loop air gap into your washer. The solution is to remove the "J" bend and snake the
Line from the trap tail. (see image) . You will run into a 90 about 8 inches in but once past that you will only have to put out 6 feet more to reach the stack. Good luck, tom