If you use the crushed ice feature frequently, you could have ice shavings building up down in the chute. I recommend that you periodically remove the ice bucket and clean out all the ice shavings down in the chute. When there is a large build-up of ice shavings in the chute, they will begin to melt and then refreeze and can cause ice to stick and jam up in the chute.
Make sure that the flapper door is closing completely after dispensing ice. If the door is not shutting or sealing correctly, warm moist air will migrate up the chute and cause condensation and freeze up in the chute and cause ice cubes to stick.
The following link of a previous answer describes the process for accessing the internal parts of the ice (and water) dispenser:
How do I fix the ice dispenser door on my Kenmore refrigerator?
If you suspect that you have broken parts in the ice dispenser, use the above procedure to access and examine the parts. If you need to replace parts, you can order them from the Sears PartsDirect website [searspartsdirect.com
If you need more help with this problem, resubmit your question with additional details.