I recently bought a house in a rainforest and the garden is almost completely covered in morning glory. It pulls off relatively easily but I'm conscious that I'm not getting the roots and it will only grow back. Is there any way I can get rid of it without fire bombing the whole area?
If you mean the wild type of morning glory (aka, bindweed) you are definitely in for a battle. However, I have had success in the past by removing as many of the plants and their roots as I could, then covering the ground with a dark mulch (many layers of newspaper, cardboard, or a good thick layer or two of heavy landscaping fabric) and after that, piling several inches of mulch on top. You can dig holes and plant a garden into this, but be aware that every place you punch a hole to plant, the bindweed will come up, climb your plants, and attempt to blanket the area once again.
It can take years to knock bindweed back this way. There may be chemical controls that will knock it back (the roots go very deep, so even chemicals may not get it all) but I haven't used those because most of my experience has been with it in the vegetable garden.
I've used Grazon (trichlopyr but DON'T use Tordon as it will kill the other plants too) - in the past to kill roots by putting the mix in a container, cutting off the plant about a foot above the ground and feeding the cut end into the container - let it have a good drink, do probably twice for each square metre.
If you spray it I wouldn't advise Roundup as it messes with soil, especially phosphate chemistry and can make growing other things difficult. Instead use Grazon and some detergent to break down the leaf waxes to help herbicide uptake.
If you have time, wait and see if the bindweed dies back a bit at the end of the season (assuming it's not evergreen where you are) - some herbicides can be carried back into the roots as a plant dies back for the season.