I moved into a new house and got into an awkward situation with my neighbour
It couldn't be too bad! What made it awkward?
I accidentally locked myself out of my apartment
That's okay. Neighbors are likely to help in a situation like that. It is also a good icebreaker.
I accidentally made eye contact and I feel awkward now.
Don't be too nervous, eye contact is completely normal! They were probably just concerned for you, and it's a wonderful introduction to who they are. I hope you can use this experience to bond with them.