The short answer is no.
Every wine contains sulfites. There is no way to avoid the presence of sulfites because sulfur dioxide (the ‘sulfites’) is a natural by-product of the fermentation process. At a basic level, sugar plus yeasts yields ethyl alcohol, carbon dioxide, sulfur dioxide and a whole bunch of other stuff that affects the flavor, aroma and structure of the wine (such as esters, anthocyanins, acetaldehydes, blah, blah, blah).
The sulfur dioxide that's produced is an anti-bacterial agent and functions as a naturally occurring preservative. That way your wine doesn’t turn into vinegar or something gross. It's nature's way of protecting the yummy wine we wanna drink!
But the naturally occurring sulfur dioxide is typically produced in really small amounts. This means it doesn’t offer long term protection for the wine. So if a winemaker wants to make a wine that will age for years or be able to sit on a store shelf, then the addition of more sulfur dioxide or some kind of preservative is necessary. And this is where you may find some options as the consumer - with the ADDITION of sulfites.
There are legally defined limits on how much sulfur dioxide can be added to a wine and overall these limits are still pretty low. It’s just enough to offer extra protection to the wine without being harmful to humans.
So if anyone tries to sell you a sulfite free wine, definitely give them the raised eyebrow. And if you see it on a wine label, then know that it’s a bit misleading. A more accurate term is 'no added sulfites'. This means there are naturally occurring sulfites in the wine but nothing extra. Keep in mind though - if sulfites are not being used to preserve the wine, something else is. And in most cases, whatever it is won't be indicated on the label. You can always contact the winery directly to find out.
Another term you might see is 'low-sulfites'. This means sulfites have been added to preserve the wine but in quantities far less than the legally allowed limit. What 'low' means is subjective though since there isn't a standard definition of the term low sulfites. If you're lucky, the label may state what the sulfite levels are so you can judge for yourself. But if not, you can always contact the winery and ask.
The best approach is to seek out wines with minimal intervention made by small producers who have the ability to dedicate themselves to the vineyard. As they say the best winemakers are farmers first - meaning they focus on growing quality grapes so that they don't have to intervene much during the wine making process. A lot of people assume sulfites are causing their reaction to a wine when in reality, the source of their issues is the addition of extensive amounts of additives during wine making.