Hi, Stephen,
It is possible to support 1.8V power rail.
As you know that the LPC8xx supports 1.8V power rail, on your target, your LPC8xx processor is powered by 1.8V power rail, of course, the SWD signals are also 1.8V power rail.
As the following figure of part of SWD connector on the MCU link board, in default pin 1&2 are populated in default for SJ1, you have to remove the resistor on the pin1&2 of SJ1, and populate a 0 ohm resistor on the pin2&3 of SJ1. On the 10 pin connector on your target board of LPC8xx, you have to connect 1.8V power supply to pin1 of connector so that the buffer circuit on the MCU link board can get 1.8V power from your target board.
In conclusion, pls connect 1.8V power supply to pin1 of the 10 pin connector on your LPC8xx target board. Pls remove the resistor on pin1&2 for SJ1 on the MCU LINK board, and connector a 0 ohm resistor on pin2&3 for SJ1.
Note that the SJ1 is closed to the LED1 position on the top right of the MCU LINK board, it is small, difficult to remove or populate any resistor.

Hope it can help you
BR
XiangJun Rong