Can underarm hair laser removal and underarm whitening be done at the same time?

There are many factors in this question. The most important one is who is doing the laser hair removal treatment. I cannot stress this enough and it is by far the most critical.  
For optimal results I recommend that you see a board-certified dermatologist who does these procedures in their clinic.
There are many different types of lasers and ipl device and bleaching creams. The optimal response and treatment also depends on your skin type and reactivity. As mentioned above make sure you see an expert.

Hi, laser for hair removal works based on the colour. So if you do with Alma Laser, it will lighten the skin at the same time. IPL have higher risk of burning skin, normally doctor will see your skin type first before deciding to proceed with IPL.