Hi there,
I am an expert Full Stack developer. After careefully reviewing your project requirements, here is my proposal:
1. I will design a App Script for your said Google Sheet file that is tracking the list of subscribers; it will have a trigger installed on it that notifies whenever the document is edited; using a simple cache of the last edited row, the system will determine if a new user email and details were added.
2. If new details were added, an HTTP request will be sent to the aweber REST API to subscribe the user. For this, the account ID and list ID will be required from you.
The script will be limited to a 30 second timeout and will also be subjected to rate limits for app script. The budget is slightly low for what you are asking for. I can develop the entire solution in two days and set it up for you. I will be happy if you can consider a budget of $60 for the project. Hope to hear from you soon. Thanks for your time and consideration.