>>889904OP, this is very feasible, but may require a few tools to be purchased.
Step 1 is to get a plank out of the log. Figure out how thick you want the paddle, add some to it, then cut that out. You can probably get 2 paddle sized planks out of the middle. This can be done with a chainsaw or even a regular ol' handsaw.
Step 2. Let the planks dry about 1 year. Set them on sticks, giving enough support that they don't bow under their own weight.
Step 3. Shaping. Did you buy any tools over the course of the year? A jigsaw is a good diy tool to have around, but you can get the general shape with the handsaw again. Just cut out the outline of the paddle.
Step 3.5 Drill any holes you want. A cordless drill is a good investment if you don't already have one.
Step 4. More shaping and smoothing. Sandpaper. Fucktons of sandpaper. Start with a low, coarse grit like 60 and grind down to final thickness and shape. Work your way up the grits once you have the profile you want. Say 120, 180, and 220. You can go as high as you want.
It'll take time, but it's nothing you can't handle, OP