Is it possible that the enzymes in our bodies use quantum tunneling?

Yes it is possible

Explanation:

These are the workhorses of the living world, speeding up chemical reactions so that processes that would otherwise take thousands of years happen inside living cells in seconds. How they achieve this speed-up - often more than a trillion-fold - has long been an enigma. But now, research by Judith Klinman at the University of California, Berkeley and Nigel Scrutton at the University of Manchester (among others) has shown that enzymes can employ a weird quantum trick called tunnelling. Simply put, the enzyme encourages a process whereby electrons and protons vanish from one position in a biochemical and instantly rematerialise in another, without visiting any of the in-between places - a kind of teleportation.