Although they are not all that common, there have been foreign monks in Thailand for decades. Some have risen to become abbots of temple and have been highly revered.
Since yours truly has never been a monk, I posed this question to an American friend who spent three months as a monk in Northern Thailand. Below is what he had to say on the topic.
"The decision to accept a new monk or not lies with the abbot of each
temple. They can make their decision using only their own consciousness
as a guide. That usually means that you must meet with them and discuss
your desire to be a monk.
In most Thai wats (temples) only the Thai language is spoken, so unless you speak Thai well it would be difficult or impossible for you to be a monk at most temples.
You will need to find a temple
where English-speaking monks reside. Also, many of the "forest" or "meditation" wats do not accept people for one week or 1 month or 3 months - they are looking for a commitment of several years.
Finally, there is no cost per se to ordain as a monk, as Buddhism is taught for free, and when you are a monk you live off the generosity of the people in the area. I do think that any temple would be pleased to have a donation from you :>)
When you ordain, the temple becomes responsible for you. If you get sick, they will try and obtain health assistance for you.
The quality of the health care depends on many things: what care is available in the local area, the type of health problem, how much money the temple may have, and (quite frankly) how willing the abbot is to spend the temple's money.
In my case, I was interviewed by the abbot and specifically questioned about the state of my health. Also, part of the ordination includes questions about your health, as the temple is not intended to be refuge for the ill. While the health questions are quite specific, they are also outdated, as they refer to common illnesses at the time of Buddha.
I took them in the larger sense rather than the specific sense, but since I was healthy, it did not matter. I did have some chronic conditions for which I already had a several months supply of medicine, so I presented no burden to the wat.
When you ordain, you must have a valid passport and visa. The temple will help you obtain visa extensions, but these are granted by the central Buddhism authorities in Bangkok and will take some time. If you have the correct paperwork, including a letter from your abbot, you will be given a temporary extension until the approval arrives from Bangkok.
As regards procedures, the basic ordination ceremony is the same at all temples, except for minor differences between the two main Buddhist sects found in Thailand. My ordination was done in two steps. First I became a "samanayn" or novice in a group ceremony; the next day I became a monk in an individual ceremony.
There are a couple of books published by Asia Books that describe the monkhood from the point of view of a farang (Westerner). One is written by a Christian author who lives in Bangkok; he is frank about not being a Buddhist but he has many Buddhist friends and his descriptions seemed quite accurate to me.
A second is by a Westerner who still is a monk. That book gives a much more personal point of view, and again the descriptions seemed very accurate to me. The second book is titled "Phra Farang", but the title of the first escapes me at the moment."