No you can't power the RS232 chip from an IO pin. You can try, but the board has a lot of capacitance and the chip has a charge pump that takes short spiky pulses of curren to generate the RS232 supplies, and any DC load and AC switching on the output will consume more current. I certainly not recommend it.
The MAX3232 and ADM3232 chips themselves work at 3.3V and 5V. The external components of the charge pump are chosen to best suit MAX3232 in 3.3V usage, it can work at 5V but the capacitors values do not match the recommended value for 3.3V and 5V operation. The ADM3232 is mainly specificed at 3.3V operation even if it supports 5V.
The MCU RX pin is 5V tolerant, and at least the MAX3232 chip supports 3.3V logic on input, so it should work.
Then, there are more downsides. If it really is ADM3232, it has internal pull-up on logic input, but the MAX3232 and other clones of it does not have internal pull-ups. Which means the PMOD design leaves them floating, and people here have asked why did their 3232 damaged and one common thing was at least lack of pull-up resistors on logic inputs. I see why they had to change the original MAX3232 chip to ADM3232, it's due to the PMOD design error of having no pull-up resistors, a common error since the 5V MAX232 and clones do have internal pull-ups. Actually, a cool niché market, if you have a board and you made the error, you can buy those chips and can't use other chips.
One final thing is that while those MCU pins are 5V tolerant, they are only 5V tolerant when the MCU is powered up, and maximum allowed voltage is 4V when MCU is unpowered. This means that if you have a 5V supply for the PMOD and you are making the 3.3V for MCU from that 5V, during start-up the 5V may be present while 3.3V still is not.
So, to be fair, you could do a small board with 5V to 3.3V regulator between PMOD and MCU boards and power the PMOD from that. It will then work at 3.3V like the MCU, have enough current, and it never exceeds 4V limit during startup.
Or rework/modify the MCU board to output 3.3V instead of 5V to PMOD.