Start GTA in Online mode

To answer the question, no. There is no way. The best way would be to just press X (A on XBox) when the game starts loading. If you accidentally load into story mode, you can use the Switcher and choose the bottom character to switch to your online character instead of having to go through the menu.

I haven't checked on my PS3, but on the PS4 you hit Options (start button on PS3), go over to Settings, down to Saving And Startup, and change Startup Flow from "Load into Story Mode" to "Load into GTA Online".

This doesn't seem to be permanent, as I have to do it every other month or so, but it beats hitting X every time you turn the game on.