Hi Jason and welcome to the forum.
I wonder if you might have an old Hobie Mirage (now called the Classic) without a plug or a drive. If so, the plugs aren't too expensive.
Even without the plug, you can certainly paddle -- the open hole will not fill your kayak up with water. If you want, you can put a little duct tape on the outside bottom to cover the opening so water doesn't splash in as you paddle. If you do, leave a little gap so the cockpit can drain!
If you have trouble paddling straight with the rudder down, try it with the rudder up; ithe boat will wiggle more but is more correctable with the paddle. Different people have different preferences.
If you have any more questions, you're welcome to post them here.