Can you put the cap back on after using each lancet? Note that resheathing is ONLY for your own lancets, not if they have been used by anyone else - get them to do their own.
Until you work out what your local medical waste situation is (you'll have to contact your local council), you can store used lancets in a plastic tub, ideally one with a child-proof top eg bleach or medicine container.
There is a lot of debate about how often to change lancets (and needles) and how to dispose of them (although options depend on local waste policies) - use the search facility to find other threads.
Clipping ends with a BD Safe Clip works for needles, but not lancets.