Yes, this will work. The Simpex VT531 has an optical slave feature — it senses the pulse of light from your camera's flash, and quickly fires its own. Note that it has two modes, and you'll need to use the right one.

S1 triggers whenever it sees any pulse. Some older optical-trigger flashes (and many studio strobes) might only have this type of mode, and in that case, you would be out of luck, because most modern automatic flashes (both for use on the hotshoe or built-in) use a "preflash" — that is, send a quick burst of light before the exposure, and use that to judge automatically how much flash to use. This includes the one on your RX100 (and it preflashes even if if the camera is in manual exposure mode; on some cameras, that's a workaround, but not the RX100).

Fortunately, your flash also has a mode which ignores the first pulse and fires with a delay. This is the S2 mode (n.b. different flash models may use different names or labels for these modes), and if you use that, you should be all set.

If you want to suppress the light from the popup flash so you're just getting nicer off-camera light, you could try covering the popup flash with a bit of opaque but infrared-transparent plastic or glass — it is usually the case that the remote flash is sensitive to IR, so you can block the visible light but still let the signal through. This may, however, mess up your cameras' metering, so you'll have to experiment and either add compensation or use manual exposure.