Short Answer is: Yes you can.
There is nothing that technically prevents you from being a 1SG with a “WALK” profile. I experienced this issue on active duty with an individual who had a “Walk” profile and wanted to be a 1SG. The issue I had with placing him in the position was that he could not meet the requirements of the physical standards inherent with the duties of a 1SG. He could not make or lead unit runs. He could not meet the overall physical requirements required of a 1SG under adverse field conditions. Therefore I could not recommend the Leader for the position. The individual was a great Soldier and Leader he just was not able to lead by example because of the profile.
There were a couple of options: