Can you make fermented, oil-stored Giardiniera at home?
If it's pickled properly (i.e. a successful lacto-fermentation, in this case) it's no longer low acid (you could even check the pH post-pickling to be sure.) It's been acidified (kinda the whole point of pickling...)

So, mixing raw (as in not processed) vegetables and oil is not the same as mixing pickled (but not cooked) vegetables and oil. Or that's the way I see it...but I'd probably still leave it in the fridge once oil got involved.

Edit: but I'd not be too concerned about shelf-life in the fridge - as an acid (not supporting botulism) food similar to sauerkraut, refrigerated life should approach "indefinite."