Hello,
As for the ability to provide electrical current,
If you look on the internet, you will notice that there are few people talking about an exact amount of electrical current that can be consumed from the board of any Arduino, unless it is a similar board that should have already been made to meet some specific requirement.
You can look at the layout of the Arduino board, and also look at the component on the board, then find the datasheet of the component and analyze the value that the controller can provide, but do not forget that the Arduino board does not have a heatsink, and this further limits the ability of the Arduino board to provide a lot of electrical current.
The most recommended is to use dedicated power for special modules, so as not to compromise the control board, the Arduino. I found some references about 200mA of consumption, so Arduino already have a good consumption being made, and if you use the USB port can be even more complicated, because the USB port should provide at most 500mA.
Mega2560 power consumption
Looking at the datasheet of your display, we can see that it consumes up to 330mA, so I would use a separate power supply, remember that this is a "luxury display", if you need an economical display there are LCD type without backlighting.
One thing that worried me is the controller model, which is the HD66712, I found reference to it being "almost" compatible with hd44780 (near 100% compatible with HD 44780), which is the controller used in the common LCD displays, 16x2 , 20x2 etc. (note: hd44780 is described as obsolete). So some difficulty may arise, it may be good to test in practice, or to search more.
Display Controller
