Of course you can.  You just have to be aware that there are costs to your health, and that you will have to either supplement with products that may be animal derived, or consider semi veganism or vegetarianism.

 

You ingest microscopic animals every time you swallow, so you may as well be just vegetarian or semi vegetarian.

 

It is a bad idea however, to enforce such a lifestyle on children.  If you don't have the time or means to breast feed them until they're 6 or 7 years old, you need to let them have milk and eggs.  To be ethical, you may have to work a little harder to make sure you get good safe raw or pasturized but non homogenized milk, and farm fresh eggs from truly free range chickens or geese, but hey, we all ultimately do.

 

If the problem you have is with killing animals that you can see, you may want to consider road kill.  Hey, it's not a joke.  Some people do actually go out and gather animals that have been freshly killed by passing cars as an ethical way of acquiring meat.