Can you use a flash drive with time machine?

A poor idea a the best, please click the link above for suggestions. What I and many other experienced users use for a Time Machine drive is the OWC Mercury Elite Pro due their superior cooling (electronics hate heat), reliability, vendor service and reasonable pricing. If you are outside the USA (OWC is a US company) use the following link to find a distributor OWC International Resellers

As the others have mentioned it is a terrible idea for the following reasons:

• Most USB sticks are extremely slow (did I mention they are extremely slow?). Any USB sticks that are actually fast are really expensive. It is much cheaper to buy a USB3 hard drive with lots more storage. A USB3 USB stick does not mean it will be fast (it has the potential to be faster, but the majority of times it is nearly as slow as a USB2 stick).
• Most USB sticks don't have a lot of storage space. Even if they do the same thing applies that I mentioned on the previous point (hard drives are less expensive per GB).
• The quality of USB sticks is extremely poor. Whenever I've performed any data integrity checks on a USB stick the majority of times at least one bit of data has been corrupted (usually multiple bits and in an extremely short period of time). While the same thing can happen with a USB hard drive, it is less likely to happen as hard drives have much better error correction built into the drives to prevent data corruption.

That would be a very bad idea, but it should be possible.

FWIW, I agree with Tim, bad idea.