You can, though it's very small relative to other characters. The way you do it is sort of hard to explain so I recommend watching video tutorials if you can't have anyone show you. That being said I'll try to tell you.
Two different ways.
Cheap way - Hold A(for multi moonwalks [I use my index finger}, Smash the movement stick in one direction (the direction you want to face but not move) then roll the movement stick down fast. For multi-moonwalks tap c-stick in the way you're facing if you're holding A.
Pro way (for single moon walks or over achievers like me)- Smash the movement stick in one direction (the direction you want to face but not move) then roll the movement stick down fast. Repeat for multi moonwalks.
Since moonwalking with Tink doesn't do too much if anything other than help with ledge grabs it's mainly just really flashy between stocks
Hope that helps, but probably not lol! Good luck with it though, it's super SEXAY!