Send file from Arduino to Server

I am currently using an Arduino Uno to log data from a push button input. Every time the push button is pressed, the Arduino logs a timestamp to an SD card locally connected to the board. Currently, I have to remove the SD card and pull the data to my PC manually. What I would like to do is, everyday at say 12:00 midnight, send the file on the SD to an internal network server.

I do not have any wifi or ethernet functionality in my system, but I have come across a number of posts discussing the use of an ESP8266 Wi-Fi Module. Before purchasing hardware and going through the hassle of programming/troubleshooting I would like some feedback on the feasibility of this idea.

My question, is this desired process of sending a file from the SD card to an internal network server possible, and if so, what hardware is needed to accomplish this task?

Thanks.

You need some form of network interface indeed. The ESP8266 is one option, and in that case you can probably even do without the SD card - use the built-in Flash storage, total 4 MB on the NodeMCU or WeMOS boards, of which up to 3 MB configurable as file storage space.

Glad to hear I am on the right track. Do you have any suggestions for how to accomplish the file transfer or tutorials related to this topic? I have yet to find a post that covers file transfer. I can only seem to find systems for transferring single data points at a set interval.

Also, the idea of simplifying the system with no SD card is tempting. However, I like the redundancy and physical nature of having the data on a separate component in case the board itself fails.