Message: what are the active chemicals in disperse dyes? what is the chemical that actually turns into a gass when heated

The answer to your question is - the disperse dye itself. The individual disperse dye is both the active chemical and the substance which is vaporized to condense upon the polyester fiber.

Do a web search for each disperse dye of interest to you, or, better, request that the manufacturer of the disperse dye you buy tell you the full chemical name (if this is not proprietary information, as is generally the case for in-house color mixtures). For example, C. I. Disperse Red 60 has the chemical name of 1-amino-4-hydroxy-2-phenoxy-9,10-anthracenedione. Each color has a different chemical structure. Some different chemical structures produce similar colors, however.

Some disperse dyes are considered carcinogenic. Others are not.