It depends on the wood chipper but none that I have yet to see will make anything like "shavings". They will produce something more like mulch, basically non-absorbant, and some of them produce more like hogfuel i.e. largely sticks and spikes and splits of fairly large size. At the very least, you'd want to SEE exactly what a particular model produces, before buying.
Second, are these live or dead trees. If they are dead, merely drying them somewhere out of the rain for a while will suffice. But if they are live trees then they will mold QUITE BADLY if you pile the stuff, indoors or out. You either have to let them compost for six months or so then re-dry them out somewhere, or I suppose if you had a lumber drying kiln-type shed you could use that but realistically nobody does
Honestly, you will get mulch that you can use for the chicken run if you use mulch there, and you can put the finer stuff inside the coop *if* you go to considerable pains to dry it and keep it mold-free, BUT it will not be absorbant or easy to work with like shavings are.
So, you can try it but should be aware that it may turn out rather differently than you're imagining.
Also chippers that do a decent job are expensive, and you could buy a LOT of shavings (especially if you can find a good source for them by the dumpload) for that price.
Just some things to think about anyhow, good luck, have fun,
Pat