Take all the current litter out of the box. Give the box a thorough cleaning, then run some diluted bleach around it. Drain the litterbox and let it air-dry. When the box is completely dry, put it down in it's usual place and put the new "litter" in the box.
If you have multiple cats, you'll need to arrange something where the sick cat is the only one who can access that litterbox (and that the sick cat can't access a second one, either).
Check with the vet; many times you can refrigerate a urine sample overnight if necessary, but it may depend on what time the lab stops by their office to pick up samples.
If all else fails, you can bring the cat to the vet for them to collect a urine sample. This may entail then keeping the cat in a cage and periodically checking the litter there; or (more likely) the vet will insert a needle into the cat's bladder and draw a sample directly. For obvious reasons, cats tend to dislike this second option.
