Good idea about the RPi!
I have a number of them running here at home and I also have an Ubuntu Server 16.04-3 running Apache (it is used for two purposes: OpenVPN server to reach my home LAN from outside and as a backup SVN server via svnsync).
I could use either of these as a middleman for the data to be sent to the external webserver.
The reason I send data to the webserver is that I have rebuilt the roof of my house and I want to monitor temp/humidity up there. So I have created his ESP-07S based monitor which can handle up to 4 different AM2302 sensors.
The sensors are in place and the controller is built and programmed. But unfortunately I fell off a ladder 6 weeks ago while inspecting the roof construction progress so now I cannot climb up and make the final connections...
I hope I will be OK in a month or so. Meanwhile the controller sits on my desk and monitors the study temperature....