My basement is flooding from my window well. There is no drain in the bottom of the window well, so it basically fills up like a fish bowl and floods my basement through my basement window. The HOA will not fix the window well, so I was wondering if I am able to install an outside sump pump to pump the water out of the window well when it is flooding. The window well is about 5ft deep, so it holds a very large amount of water. I have already replaced the window a couple of times as it breaks due to the water pressure against it.

Sure. Hope the HOA doesn't object. Where will you pump the water?

Are you in an apartment, a Condo? Townhouse, single family home? In most cases its not the responsibility of the HOA to fix that... but the property owner. Except in the case of a condo or Apartment where the outside walls are a common property.

You will need the permission of the HOA since the well is not your property. If the well fills with water you also have a grading problem which could be corrected by the HOA. Who denied your fix from the HOA. A general manager or the board of directors? Eep appealing that request since it really is their problem causing you to flood.

And if you are in a cold climate, imagine a foot or more of wet snow in that opening, freezing into a solid block of ice around the sump pump at night and staying that way for a few weeks, followed by heavy rains. We had mounds of ice that lasted for more thana month in my area this winter. Treat the disease (one possibility insufficient slope away from the building) instead of the symptom. Have you tried a plexiglass window well cover (readily available at hardware dealers of all types) sealed down with good outdoor caulking to try and minimize the problem?
Also, could this be a rising water table rather than surface runoff? Sounds like that is a possibility if the ponded water breaks the window. If so, you have a much more serious situation.