Can't see any reason why eucalyptus bark would cause any problems. It's pretty inert stuff

There is still a question as to whether eucalyptus bark has toxins in it - I've seen for and against's for using...

It's possible that the bark has mild allelopathic properties, but if in doubt, let it rest in a pile for a year (it takes ages to break down, so will be usable for a long while yet), and then spread it about.

I certainly wouldn't use it fresh - eucaplyptus growers do not recommend it as a mulch, and any fresh, uncomposted chipped bark laid onto soil will take up a lot of nitrogen as part of its decomposition process, and that's nitrogen your plants require for growth. Best composted for at least a year if you're going to use it at all, but the nursery growers of this plant recommend its use for pathways on open soil rather than as a mulch around plants, although it is certainly for sale, bagged, as a mulch.

I have many eucalyptus trees near my vegetable garden, is it possible they are effecting the growth of my vegetables? They seem stunted and are not producing fruit.

Studies have shown that some eucalyptus varieties have an allelopathic effect on some crops. You could research whether what crops you're growing are affected by whatever varieties of eucalyptus you have, the info is available on the internet, although not all food plants have been tested against every variant of eucalyptus, so it might not be that easy to find out.