Yes, as mentioned above, you can work in Thailand without a work permit, but it is illegal and penalties are fairly harsh. I wouldn't take any chances in Phuket, as when I was visiting there I was told by other instructors that the Immigration (or was it Labor?) Department had its enforcement people at the pier keeping an eye on divers, and one instructor who was in the process of getting her work permit (but didn't have it yet) got busted, went to jail for a little while, then got fined and deported and is barred from reentering the country. Not good.
I've heard of little to no enforcement on Koh Tao, rare but occasional enforcement on Koh Samui, and I don't know about elsewhere.
Your employer can sponsor you for a work permit, but can only do so for 2 employees per 2 million baht of registered company capitalization. Many dive centers keep their registered capitalization level low -- either because they don't have much capital, or because if it's below a certain level they can avoid the hefty VAT taxes the bigger companies must pay. The long and the short of it is that you're not very likely to be sponsored by your employer until you've been here in Thailand long enough for a dive center to be sure it wants to employ you full-time for a lengthy period (for example, you've been working for that dive center as a freelancer for a while and they trust that they can rely upon you). So, many instructors here have thus gotten their own work permits without any assistance from their employers.
The way to do that is to set up a Thai company here (with majority Thai shareholders), and the company then sponsors the instructor's work permit. It can be costly, so you'll want to be sure you want to live and work in Thailand for a long enough time to justify the expense. As someone else commented, you should get a multiple entry, non-immigrant B visa before you come to Thailand, as you'll need that visa to apply for a work permit.