For golfers, grabbing your putter as you walk to the green is an instinct. You putt with your putter, duh. But, believe it or not, the Rules of Golf do not require a golfer to use a putter on the putting green. In fact, a golfer can use any club they want on the putting green.
Under the Rules of Golf, a golfer can use a putter, a wedge, an iron, a fairway wood, a hybrid, a driver -- basically, whatever of their 14 clubs are in the bag -- on the putting surface. You've probably seen a number of professional golfers use a wedge or a long iron as a putter in the event they break their putter or it is otherwise damaged during a round of golf through unnatural means.
The natural follow-up question, of course, is: Does a golfer have to make a putting stroke using any club they use as their putter?
No, they don't.
A golfer can use a wedge to hit a chip shot on a green if they feel it will get them closer to the hole than putting the ball on the putting surface. An example of such a situation might be when a golfer's ball would have to travel off the green -- through the fringe, for instance -- for it to get to the hole with a putter. In that case, a golfer may use a wedge to chip the ball from one part of the green to another, going over the obstructing portion of the green, for the best result.
Of course, most golf courses would frown on this, particularly if a golfer takes a big, fat divot with their evasive chip shot from the green. However, in the heat of competition, there's sometimes no choice but to hit a chip shot on the green and tell the grounds crew "Sorry!" for using a wedge.