The Sony DSC-RX100M3 III does not have a built in hot shoe to clip on a large flash. No problem with the RX100m III as all you will need is a slave capable speed flash.
Since the newer Sony RX100m III has a built in pop up flash you will be able to use the camera’s flash to trigger a larger and brighter external flash.
The speed flash you buy must have the option to be used as a slave flash that can be triggered from another flash source. You can also buy a simple flash sensor trigger if you already own a simple speed flash without the light trigger.
The flash I have had great success with when using in combination with the RX100m III is the Neewer TT560 Flash Speedlite (The light in the top image). It is almost always triggered by my little Sony camera.
If you want to use the speed light at a greater distance for the camera you can place a small piece of aluminum foil on the pop up to angle the cameras flash towards the speed light to help trigger it. You can also use the foil to limit the cameras front firing light and only get the off angle flash from your speed light.
Using the speed light to bounce off a white ceiling will greatly help fill a dimly lit room that your built in flash could never do. Adding an expensive speed light to your camera bag is one of the best first purchases you can make for your Sony RX100m III.
Links to speed flashes and items in video.
Light I use and trust to work – Neewer TT560 Flash Speedlite
The flash sensor trigger cubes – Flash Slave Trigger