It's definitely possible - you can also partially form it and then have the Duty Finder fill in the empty spots in the party by registering your partially formed party for duty, although this can take a while. But more to the point, you can definitely preform your own party and enter the dungeon yourself (either through the Duty Finder or by clicking on the 'entrance' target at the dungeon's front door.)

I've heard preforming your party allows you to go in without requiring the 'holy trinity' of a tank, healer and two DDs, but I'm not sure if that's true or not.