Well, the question is crystal clear. How do you ask someone (politely) to clean up the room using a vacuum cleaner?
"Politely" has various levels so I'll try to address several:
Would you mind if I asked you to hoover (vacuum) the room?
This is probably very British. It's very polite. Notice we haven't actually asked for the room to be cleaned, we have asked if it's a problem. If the person you're asking is British, this is the most polite method. You can even experiment with words like "awfully" (Would you mind awfully...?), or by adding other conditionals (...if you have time).
Do you mind hoovering (vacuuming) the room?
Would you be able to clean the room, please?
Are my firm favourites for sounding a) polite and b) not too polite as to sound foolish.
Lastly, I offer a more familiar (though polite) option (assume you have known the person a while but in a respectful/businesslike way):
Is it OK if you vacuum the room, please?
Could you vacuum the room please?
Hoover = British English
Vacuum = American English
If this is based on a real situation, it might not be as simple as "being polite". The issue could overlap into the realm of interpersonal relationships and the use of higher EQ (emotional quotient; dealing with people in an emotionally savvy manner).
For example, suppose it's a roommate who is messy and you want them to vacuum your shared room or common room. In that case, even a polite request to clean up could cause problems. It may be better to write up an agreement on ground rules, who does what cleaning when, etc. Checklists could help.