Vacuum is the correct word (verb).
Hoover is a brand of vacuum cleaners, but it has been integrated into the language to the point that lots of us call it ‘hoovering’.
In context, doing the ‘hoovering’ is the same as if we said ‘I’m going to do some dysoning’ sounds a bit silly, and that’s why some people have a bee in their bonnet about people saying hoovering.
Did you know the word “Hoover” is now in the dictionary and means to vacuum.