The idea of "conquering" or formally incorporating territory wasn't held by the rulers of Siam/Thailand until the mid-to-late nineteenth century, when it hurried to reinvent itself as a modern nation state in response to colonial pressures from Britain and France. By then, Britain had already formed protectorates over the southern Malay states. Siam did attempt to extend control and implement more centralising reforms over the northern Malay sultanates, which had been in a tributary relationship with Siam, but ultimately ceded Kelantan, Trengganu, Kedah and Perlis to Britain in 1909. Pattani, however, was assimilated.

Earlier, though, in the 15th century, Ayutthaya-based Siam appears to have held overlord status over much of the Malay Peninsula, at least to some degree and for some time. There are records (though not undisputed) that Singapura and Melaka were tributary to Siam, and Pahang even earlier so.

As for the part concerning Myanmar, Ayutthaya did control the western upper Malay Peninsula for the majority of the period from the late 15th to the first half of the 18th century, enabling it to profit from the trade routes linking the Indian and Pacific Oceans. Siam was unable to retake the coast after the fall of Ayutthaya, though, and it was the British who again, having conquered Burma, finalised Siam's modern borders with it.

So the short answer to the question would be, though "conquer" probably isn't the right word, most of the Malay Peninsula did at some time come under Siam's sphere of influence.