If you use YA's advanced search option for the words "Peace Corps" you'll get some sponsored links. Many will expect you (or your friends and neighbors) to bear some or all of the cost.
The Peace Corps gives you three months of training. You'll learn your host country's language, history, religion(s) and culture, plus get some vocational training. (I learned Malay, Iban and Hokkien, then practice taught for six weeks, under the supervision of a master teacher, for instance.) They provide dental and medical care, a living allowance and some after-service placement counsling. They pay for your air fare over and back. It would not be cost-effective to do all that for someone who stayed on the job for a couple of weeks.
If you want to pay your own way, provide your own medical / dental insurance and already speak the language, there are a number of organizations that will place you for as little as a week. Some people take two-week "working" vacations as volunteers and have a ball. Some don't even require you to speak the language. (Not a problem if you volunteer inside the USA).
Here are two:
(Based in the UK, takes volunteers from anywhere, has a six-month "programme" for people 18 - 25) I knew VSO volunteers when I was in the Peace Corps 30 years ago. They are a solid, well-known organization.
Student Conservation Association
My daughter spent a summer with them. You pay for your food and air fare, they loan you a tent. You spend 4-6 weeks doing manual labor in a national park in the USA with a great bunch of other kids. They will take volunteers as young as 16.
Here are some others have mentioned. I know nothing else about them:
This is the a long but comprhensive page about volunteering.
It has a list of reputable organizations, but you should read
the advice, too:
You didn't ask, but those 27 months in the Peace Corps may be the high point of your life. You'll certainly be seeing the world in a new light, smelling new smells, eating new foods, meeting new people. They fly by. At 22, looking forward, they seem like a long time. At 60, looking back, they don't.