Maybe, but unlikely. You can ping your router if you want to check what wireless latency you're dealing with. For example at home, I'm close to my router so if I ping it from my laptop, it's about 1ms. Basically almost the same as LAN. The LAN adapter helps because it keeps you away from other wireless network traffic, and prevents any other issues (interference, etc). Also some routers just suck with wireless, so having a LAN connection just means more stability.
I'm planning on purchasing the adapter, but I know it won't make any difference in terms of online. Why? Because it takes 1ms for my packet to get to my router through wifi, meaning if I compared results to my lan adapter it would be the same.
MK8 runs better because it's probably handled differently, Smash is Peer-to-peer so the reason you can have laggy matches is because they can be lagging, they could be very far away from you distance wise, or they have someone downloading on their connection.