Running 'git' in AWS lambda

Yep, this is definitely possible, I've created a Lambda Layer that achieves just this. No need to mess with any env variables, should work out of the box:

https://github.com/lambci/git-lambda-layer

As stated in the README, all you need to do is add a layer with the following ARN:

arn:aws:lambda:<region>:553035198032:layer:git:<version>

(replace <region> and <version>, check README for latest version)