When we mean "to try", an alternative to "to have a shot at something" is "to give something a whirl". Think of a roulette wheel. When the wheel is turned, it is "given a whirl".
I don't dance well, but if you want to dance, I will give it a whirl.
It's my first cake. Today I thought I would give baking a whirl.
I don't know if I can fix your camera, but may I give it a whirl?

