No, you get to (have to) attempt to remove Shaken only when your action comes up.
On their action, a Shaken character must attempt to recover from being Shaken…
(SW Deluxe, p. 68; see my footnote below)
This is a precondition for taking the action, and there's no other circumstance that allows making this free roll. This means you can only attempt it once, and only when you're already Shaken when your turn arrives in the turn order. If you've become Shaken as a result of your own action or others' activity during your action, you don't get an immediate free attempt to remove it.
Of course, you can spend a Bennie at any time to remove Shaken.
A player may spend a Benny at any time to remove his Shaken status. (ibid.)
- Action here almost certainly refers to your turn to act rather than individual actions. It was more explicit in SWEX and SWR because the rule used the word “turn”, but it still appears to be the intended meaning in SWD because the effect of failing the attempt dictates that no actions at all may be taken other than free actions. That also means that even if it was per action when taking multi-actions, the second and later actions would never start. So one recovery roll holds for the entire turn, any way you read it, making the “they still mean ‘turn’ in SWD” reading simpler and so most plausible.