How do you approach someone to ask for a letter of recommendation?

I would set up an in-person appointment and ask politely. Makes it harder to turn down on their part. :)

A hand typed letter outlining your exact request and reason(s) why this letter and what it would mean to furthering your pursuit of this scholarship…Hand deliver it and look him in the eye and say…Sir, I respectfully request your assistance in pursuing my dreams. I think this gentleman would find it an honor to oblige your request. Be bold and never back away from each and every opportunity to make this scholarship the best opportunity you have. Good luck

I would go and set up a meeting with the person and then in the meeting explain that you are applying for scholarships and would like to ask for a letter of recommendation from them. Usually bosses have no problem doing this if its going to be bettering you for school or something like it.

I strongly agree with @seekingwolf. Send an email to set up an appointment:

Dear Bigwig,

I’m a student in your department/took your class last spring/whatever. I’d like to meet with you briefly sometime in the next week/2 weeks/month at your convenience. It should only take ten minutes or so. When is a good time for you? Many thanks.

Best regards,
SuperMouse

When you go to the meeting, bring a copy of your transcripts and any other relevant information (cover letters, essays for the thing you’re applying for, whatever) about yourself. Also bring some information about the scholarship or program you’re applying for. If you’re feeling optimistic, include a stamped, addressed envelope.

I have to ask, though, are you sure you know these people (you are thinking of asking 2 separate people, right?) well enough that they’ll be able to write you a strong letter? There isn’t a lot of information here, but I got the impression you haven’t had much contact with them. If they just don’t know you that well, it’ll be tough for them to write you a strongly supportive letter. Not because they don’t like you, they just might not have enough to go on.

To expand a little bit on what @nikipedia said, if the person doesn’t like, respect, and know you well enough that’s it’s immediately obvious they’ll say yes if they can, then what are they recommending?

The act of asking is more about whether or not they have time and are comfortable putting the effort in to really make you shine.

Someone higher up writing that your file doesn’t show any demerits and they’ve heard good things won’t sell you as well as someone who enjoys working with you and can give impressions and examples of what makes you stand out.

The person reading it probably won’t care if it’s from a VP or the bottom man at the office, they don’t know the people, so it’s just a name.

Good luck with the scholarship.