I have a growing interest in medical anthropology and I'm taking my first steps to traveling overseas to work/volunteer/intern.
Could someone recommend avenues for making this happen? Specific organizations?
I did two trips with ISL, in Central America and in Mexico. Any similar org like CFHI, Unite for Sight, Operation Smile, etc. that lets you pay to work in a rural clinic is going to offer more or less the same experience. Going on these trips was worth it for me, because I got a teensy little taste of what MSF et al would be like, and because I get to check off the box that asks for "field experience" when I apply for an international rotation as an M4. But it grates on me that most of the money for my trip stayed in the US, and that the "work" we did was mostly for show, and that the organization was covertly religious.
What I recommend is to first think hard about what part of the world you're most likely to want to invest in. Think about language skills, government stability, longevity of aid programs, etc. Do you want to be in South America? Africa? Southeast Asia? What language(s) do you need? Where are the major aid orgs based - in Europe or in the US? Make a bet, that you're willing to lose, on what locale you want to focus on.
Then pick an org that does premed trips to that locale, and spend the $1k to $5k to go on a trip of 2 weeks or more. Use that trip to find out if you can stand heat and lack of plumbing/electricity, to find out how it feels to not speak the language, to find out whether you like working with impoverished populations, etc. Ideally, on this trip, you'll be able to set up a long term relationship with an individual or a local organization that you can come back to, on your own. In many cases, you can fundraise to help pay for such a trip, using your church or whatnot, because the orgs are usually nonprofit.
And then as soon as you have the guts and the time to do it, go back (or don't come home right away). Be there and contribute without the support of a US organization. Use the contacts you make during your organized trip to find housing and work (don't plan on getting paid). Build up your language skills. Use public transportation. Do non-medical work if that's all you can find. But
live there. Make friends; join the community; go home and raise money for a cause. Do this as often as your med school schedule will allow, and work it into a 4th year rotation in that locale's closest major hospital.
Are there any reputable organizations that pay for internships or hire employees?
Until or unless you have a skillset that is needed and unavailable in a particular locale, you're not going to get any financial help to work internationally, nor should you. Dollars that you use up are denied to the local work force, which doesn't solve any problems. Your money is the only real contribution you can make, until you have a needed skillset (like an MD, DO, RN, etc.). And as a US "voluntour," most of your money will go towards keeping you housed, safe and entertained.
You can sometimes find work teaching English. These jobs tend to be more urban, and in order to get paid, they usually want you to have a certification that is expensive and time-consuming to get. But teaching English might be one good angle on the "live there" strategy.
Also look at organizations like PIH, which occasionally hire interns or temporary workers. Right now, for instance, the job of traveling nanny for a medical director is open, which would be a freaking sweet, paid, heavy-exposure job with connections, if you have a year to blow.
Best of luck to you.