Any organization that regularly books flights has probably encountered this question before and may well have a policy to address it. Rudeness is a cultural question, and you haven't specified where you are, but it wouldn't generally be rude to ask this in the United States.
One key thing to keep in mind is that travel prices vary, often significantly, depending on the travel dates. The organization likely doesn't want to incur any additional costs for your vacation, so they may only be willing to pay for flights as if you're traveling Friday-Sunday, if the weekday flights are more expensive, with you paying the difference. That, in turn, may create accounting headaches they don't want to deal with.
Some companies do have policies against doing this (insurance/liability reasons are sometimes mentioned, or just a policy to simplify the planning and accounting). But I don't think it would be at all rude to ask as long as you are clear that you'll make it cost-neutral for them and would understand if this isn't possible.