I used to absolutely need a uke strap when I was starting out with my Kanile'a tenor, but as I have progressed as a player and have slightly better overall form, I find I am able to hold the ukulele with my right forearm.. I also sort of cradle the neck more like a guitar in my palm, and I use my thumb to mute strings and fret the G string, so that helps support the uke as well. Straps are absolutely fine, though, no problem using them fashion-wise! James Hill does use one, and he is easily one of the best ukulele players in the world.
Cheers