Might open the window as well is grammatically correct, but it is recommended to use the may tone to be more positive.
It’s OK to write “as well” at the end, but the implication here is that before opening the window, doors or other (air flow) channels have been opened, or other related actions and operations have been performed. It must meet this condition before it can be said to be as well. If it just opens the window and does no other operations, it cannot be said to be so.
Might as well open the window is another way of saying it. It is more appropriate to use it here to express "might as well". It does not require related actions as a condition and can refer to opening the window alone.
Hope it is useful to you.
Supplementary answer to the question after the original poster: If this is the case, the test is wrong and should be as well as possible.