It doesn't make much of a difference. I'd stick with a yellow onion myself as I like their flavor balance.
But it's more about the rest of the recipe itself I think.
There are onions I wouldn't use. I'm not a fan of red onions for this myself as they can show some odd colors depending on the recipe. And as this soup can easily cross the line to sweet, I'd avoid the sweet onion varieties.
Personally I use one of the sweet onions such as Vidalia. I figure they have a bit higher sugar content and should therefore caramelize better. I have also made Frencho Onion soup with red onions without problems.
By the time many onions are carmelized a lot of their individual taste characteristics are cooked away, leaving just onionness, I guess you could say.
When I make my sweet onion soup where the onions are basically poached in butter, salt and their own moisture and not allowed to brown, I do notice a difference between regular and the low sulfer sweet varieties.
Sweet onions may have a bit more sugar than regular onions, but they don't always, and they've sometimes got less. What sweet onions have is an absence (or lower levels of) of the sulphur compounds that make onions (and alliums in general) pungent. That's largely a function of where they're grown, which is why the best known variaties (Vidalia, Walla Walla, Maui) are named after particular places.
The sulphur compounds are broken down during cooking. The only thing you'll gain from them is less irritation when cutting them up, which might be enough reason to use them. It's not for me, but I'm cheap.
what said above about sweet onions is true, they usually tend to have less sugar than your regular yellow onion. They really should only be used raw IMO, thinly sliced on salads, sandwiches, etc. as they don't have the agressive bite, but you tend to get the best result with regular onions. Nice and slow, stove top or in the oven. Make lots and freeze it