Yes, it is possible to rent one (or more) nights at a DVC Reort from a member. Just respond to any of the threads on the DVC Rent/Trade Board with the dates, resort and type of villa you will need and see if that member can assist.
If you are more than 7 months from your arrival date you will need to find a member who owns at the resort you want but if 7 months or less, any member wil have access to reserve availability inventory. As with anything, the earlier you start looking, the better you chance of success.
You might also want to review the **How to use this Board** thread near the top of the Rent/Trade Board. it has some great suggestions about how to proceed.
There have been some posters trying to bypass the Rent/Trade Board by contacting others by PM or email after seeing their requests on the Rent/Trade Board and some of these transactions have not turned out well for the renter. If you are contacted about renting by anyone other than a thread to which you have responded (either by posting in it, by PM or email), please let any of the Rent/Trade moderators know.