My tip for you would be to do it while you are still in school. At least for the school I was in at umich you could do internships and get credits written off (which I did). It’s also easier to get internships abroad is you are student because there are a lot of European internships that are only open to students. Its much harder to get internships post grad because once you are no longer a student, you have to get visas under worker categories which are more competitive.
You’ll have to do the leg work to find and secure the internship though because at least in my experience, that wasn’t something umich had ties with companies abroad on. But I know people who took a semester off to do internships abroad and they really enjoyed them.
I don’t believe there is any country on the American continent that lets you stay as a visitor for a year, most are at most a few months and with stipulations (like as an American you can spend significant time in Canada as a visitor, but you might run into trouble enrolling your kids in school as they don’t have legal status as Canadian residents).
If you want to stay for a year, you will likely need something beyond a visitor visa. You will need to do more research into what the rules are in your countries of interest and what immigration status you would need to live and enroll your kids in school here.