My dog (yorkie/chihuahua mix) killed a chipmunk in our backyard.

I was outside swimming in my above ground pool. He likes to hang out outside with me. I knew he was interested in chipmunks/squirrels and has chased them (until they escaped under the fence or into a tree) but I never thought he would catch them, let alone kill them! I didn't hear a sound- I just looked over and he had it in his mouth. I said "No!" and climbed out of the pool and got over there as fast as I could, but it was too late. The chipmunk(s) in our yard have a little tunnel under our fence they use to get to the birdfeeder. My dog does spend a lot of time staring at the tunnel, I guess I should have known better.
Is this common among terriers? (My husband thinks so.) I had two chihuahuas for over a decade and they never caught, let alone killed, anything.
I guess I can't let him out of my sight in the yard now. Maybe I should use a tie-out when we're out there? I'm also worried about disease from the chipmunk. He didn't appear to break the skin, but it was as dead as a doornail when he dropped it. I'm really upset about this and feel like it's my fault. Any suggestions of what I should do? Thanks.