No you don't.
Wifi in the BD player serves three purposes.
The first is for firmware upgrades. You could periodically wire it to your router to do that.
The second is to use built in apps for streaming services. If you don't use any then it won't matter if you don't have Wifi
The third is for BD Live which lets the disc you are playing access additional material about the movie. Movie will play fine regardless.
Specific make and model would depend on what is available to you and the price range you want to go for. I don't think you would go wrong with any major brand such as Sony. Panasonic or Samsung. At a higher price Oppo is very good.