There are dedicated drawer bottom planes available- these are essentially small plow planes with an adjustable fence and a couple of irons. They are almost always British and made by the big names- Marples, Mathieson, Preston, etc. They aren't terribly common, but they aren't expensive either. Guys like Tony Murland or Patrick Leach might have these available.
By far the least expensive way to go is to use the grooving plane from a tongue and groove plane pair. These planes are available in a variety of sizes and generally cut a groove the perfect distance from the edge of a board for a drawer bottom. If you find a single plane without its mate the cost is very low. If you find the matched pair even better! Then you can cut T&G joints with style and plow those drawer bottom groves with ease. These are also really common and easy to tune.
(yay hand tools!)
Josh