Islamic marriage in Thailand does occur and many Muslim expats do get married. The biggest hurdle most times in Thailand is getting a Freedom to Marry letter from the Embassy. Many Middle Eastern Embassies require that your documents are brought from home, such as your status letter which shows that you are single. Some Embassies also will not issue the letter if you are getting married to a non-Muslim.
The first step is to perform the conversion of the non-Muslim, then proceed to have your wedding at a Mosque which is then followed by registering the marriage in Thailand. There has over the years been a growing number of converts to Islam in Thailand mainly from Southern Thailand. Many Muslim expats simply want to have an Islamic wedding without the official government registration. If however you are wanting to take your Thai wife back home then she is going to need to obtain a visa and a traditional Islamic marriage will not get this visa.
You can search this website for information on the letter of affirmation, marriage registration as well a property rights in Thailand. It is best to read all of them in one sitting to get a view of the laws in Thailand and how they are applied to marriages compared to your own home country.
Getting married as stated above requires that you contact your Embassy in Bangkok and ask what they require for the Freedom to Marry letter. If you are from Malaysia then you need a status letter from your Home Affairs Office in Malaysia. There is only one Thai law firm in Thailand that does Islamic wedding conversions as well as the marriage registration, being GAM Legal Alliance in Bangkok.
When you have the required documents then the marriage can be arranged. Should you be getting married to a non-Muslim then the conversion process needs to be completed first. This is usually done at the lawyers offices. Note that the Islamic Centre in Bangkok is very busy and you will need to make timely arrangements for this. Booking is essential for this to be done. This process takes about 30 minutes. You will note that the Imam will need a letter from her parents consenting to the conversion.
Once the conversion process has been completed you can have your traditional Islamic wedding in Thailand. Your documents after this will need to be certified by the Thai Department of Foreign Affairs in Bangkok and taken to the Amphurs Office for the marriage registration to be completed. The process can take few days depending on your nationality as each Embassy has their own process.
Now that you are married you can stay in Thailand if you meet the financial requirements as set out by the Thai government. If you are not going to live in Thailand then the attorneys can arrange for a visa for her if you are taking her back to the US on a US marriage visa for a Thai or to Australia which will require an Australian Visa for a Thai national in Bangkok.
Speak to a lawyer before you get married in Thailand to ensure that you understand the property laws in marriage as well as the marriage requirements.