I've been traveling in Bangkok, Thailand for almost a month now and while things aren't that different, there are things that are not the same as in the US.
I learned quickly to bring your own toilet paper with you when you are out in public. There are 10 baht machines inside or outside the women's restroom for a 'sanitary napkin' or toilet paper and a menstrual pad.
More established places such as ICONSIAM, Terminal 21, and cafes will have toilet paper inside the stalls. Still, you should ALWAYS throw your used sanitary napkin into the trash can. Don't worry; there is a bidet or 'bum gun' so you can clean it and then dry it off. :)
In accommodations, it is typical to have an open-air restroom. For example, a mesh screen in front of an opening to prevent insects from entering. The purpose of the open-air bathrooms is to obviously have the odor rise and exit and dry the water on the ground the shower creates.
I haven't had much luck finding tampons at 7-Eleven, Family Mart, Lawsons, or Tops. Boots sells their brand of tampons in light, regular, large, and extra-large. I opted for the compact ones as I only have a 46L backpack and one pack of overnight pads. I haven't found any thong panty liners, so if you do, let me know!