looking at the source of the page, I can see that you have changed the name of the checkbox, in power store the code to generate the checkbox is:
<input <?php if (!(strcmp($WA_Store_Cart->DisplayInfo("Quantity"),0))) {echo "checked";} ?> type="checkbox" name="WA_Store_Cart_Quantity_check_<?php echo $WA_Store_Cart->DisplayIndex; ?>" value="0" />
this will crate a checkbox like:
<input type="checkbox" value="0" name="WA_Store_Cart_Quantity_check_0"/>
yours aree being created with a different name:
<input type="checkbox" value="7" name="WA_Store_Cart_Delete_0"/>
there is also a hidden form element that should be created:
<input type="hidden" value="1" name="WA_Store_Cart_Quantity_hid_0"/>
This is missing from your cart page. It would seam you have made some significant changes to the cart display page that are causing the issue.