Yes, you can do so. The cheapest option at the moment is to use a TubeSat kit from InterOrbital Systems ( http://www.interorbital.com/Tubesat%20Kits). The kit is $8,000 and includes a launch into low Earth orbit.
You will need approval if you want to take images of the Earth. Take a look at DIY Satellite Platforms by Sandy Antunes for a detailed overview and guide.