If I put a tree in a wood chipper, can i use the chipped wood in my garden?
On top of the soil, yes.
In the soil, not such a good idea as already mentioned.
Termites- It could, depending on your area and where you plan to use them relative to the house. In other works, if you use them a good ways from the house, it could draw them there away from the house. Or it could draw them into your yard from the neighbors. This could be similar to using Japanese beetle traps.
Most chips from a tree crew chipper are coarse and irregular. They won't look good in a flower bed but if you wanted to use them in a vegetable garden and till them in at the end of the year probably so. Or if you use them around shrubs at the rear of the yard.
If it's worth doing, it's worth doing right. If you wouldn't put your name on it, it ain't done right!