Things to say when approaching hotels??
"Do you have any vacancies?"

Why don't you want it to sound like a sales pitch... it IS a sales pitch! You should be proud of yourself and confident. After all, they are a hotel... with carpets, upholstery and mattresses.... they have a need for your services.
You just have to avoid sounding like the brainless minimum wagers who make sales calls all day long that start with the line "can I speak to the person in charge of XXXXX please......" you know, the ones you hang up on.
It's difficult thinking of an opening line that isn't immediately going to activate the defences of whoever answers.
Depends what sized hotel you're phoning as to the approach. It's safe to assume that in any decent sized hotel the call will be answered by a receptionist, and the person you'll want to speak to will be a facilities manager or head housekeeper
In smaller hotels it may well be the owner who answers the phone and makes decisions on cleaning etc, so a bit of homework might be needed.
In all honesty, hotels will be a tough nut to crack. The national chains have national contracts with big companies, following pre-determined price structures and service levels... the door is likely to be firmly closed in those cases. But the smaller independent ones might be an option.
I think hotels are the kind of place you'd be best making a call in person, nice warm smile as you walk up to the reception desk and make your opening line count!