The only thing that needs support is the neck and you might be able to print it without depending on your printer. Remember to print 6 of the machine heads. You might be able to print it at lower resolution but I cant guarantee that all the tiny parts will be made properly.
Some details like the bridge casing and frets don't seem to print at this scale. All of the detail is in the model however and may print if a smaller .2 nozzle is used. You could also scale the entire model up a bit if you want.
I haven't modeled a whammy as I hardtail all my rigs :)
I printed this mostly at 40 micron to keep as much detail as I could but it could be printed at .1 or .2mm without too much issue.