no
Wipers on headlights engage when wiper on windshield engages.
There is no button to operate the headlight wipers. The headlights washers and wipers are activated along with the windshield wipers when you use the windshield washer by pulling the wiper switch towards you.
Headlights need to be used at night, when it is dark, and when windshield wipers need to be used.
Wipers
They come on when you turn on the windshield washer, but only when the headlights are turned on.
your multi function switch is more and likely going bad
Your headlights should be on whenever it will make a difference on the road. If you can look at the road and be able to tell that your headlights are on/off, then they should be on. Some states, such as New York, require headlights to be on when visibility is limited, such as in rain or fog, even in the day, regardless of whether or not you can see them on the road. In fact, one of the criteria for headlights is whether or not you are using your windshield wipers. At dusk or, in some states where mandated, when it is raining or foggy and you need to turn on your windshield wipers.
Windshield wipers don't work
Why are there windshield wipers inside the car?
that switch does not exist. It simply is activated when you spray your windshield washer fluid onto the windshield. Your headlight switch has to be on. I know your lights are always on, however the wipers for your headlights will only activated if the headlight switch is in the on position.
Your headlights should be on whenever it will make a difference on the road. If you can look at the road and be able to tell that your headlights are on/off, then they should be on. Some states, such as New York, require headlights to be on when visibility is limited, such as in rain or fog, even in the day, regardless of whether or not you can see them on the road. In fact, one of the criteria for headlights is whether or not you are using your windshield wipers. At dusk or, in some states where mandated, when it is raining or foggy and you need to turn on your windshield wipers.