OMG! I remember using a blender like that when I was a kid. I didn't know they still made them.
Just my opinion but I don't think it would produce too much air in the batch since it will be operated by muscle power. I don't think you could blend it fast enough. However, it would probably be like making whipped soap if some air is whipped in so I say give it a try.
I think it will intro air depending on the depth of your soap. If any of the beater part is above the level of the soap you will probably get air in it. But it c would also depend on how fast you operate it ...... I can get one moving fast enough to whip eggs or whatever, which is intentionally getting air into the mix. (The gearing makes them move pretty fast, Hazel)
These are made to whip eggs and other food products which adds air into the mix, so I'm sure it would do the same for soap. Not saying you can't use it, but I wouldn't expect the same result as with a stick blender.
But it wouldn't add as much air as a stick blender. Plus soap batter is a little heavier than eggs. I don't think it would be a problem because if it does add some air, it's going to be like whipped soap.
eta: I didn't see that xyxoxy had already replied.
Be careful about using a wooden spoon. Make sure you never use it again for anything but soap unless you like the taste of lye. And in time it may actually eat away the wood. I have some plastic spoons that look like wood that I use for soaping, but I avoid using wood. While we are making a more natural type of soap, my recipes anyway, are not the same as those my grandmother made, so I don't feel at all bad about using modern equipment for modern natural soap.