How much effort does it take to modify a reference letter for different applications?
If these people believe you are deserving of an award and have already written you a letter, they certainly won't mind modifying it for this purpose.  If they don't think you're deserving of an award then of course you don't want them to write you a letter for this purpose.

The answer to #1 depends on the letter writer and the specific applications/awards involved.  But it is a much smaller effort than writing the initial letter.

The answer to #2 is that yes, this is reasonable. Remember that your letter writers were once in your shoes, and asked people to write letters for them.  One day you will pay it forward.