Take great care to press them evenly as bending an axle as you do is very possible. I've used drill presses with/or a "tight vice" (alone,low slop) without bending; but I've pressed gears since I was very young (gramps made special temp/pressure gauges for chrome processing in industry...in the garage, I was his "drill press monkey" preschool and after school) Just be smooth once moving and mindful of too tight a fit to drive in fully (reamer time on non-spline. On splines maybe, but inspections and grinding options are more complex.)
Wood in the jaws 
Use the jaw center, low; as close to the vice-screw as possible. The ends and high jaw are off "true" a degree or two (more like minutes/seconds) Mind rod bosses might need a hole in the wood, some wheels badly need a wheel cup to hold them correctly.
