In theory, yes, those should work, as long as you have a hollow space behind the drywall. You should not load up the wine rack completely, though. The weight rating on those is 143 lb in 1/2" drywall; since you've got 3/8" drywall, that rating decreases to 97 lb.
The weight of the unit is 24 lb, and it says each shelf can hold 35 lb. All together, that's 94 lb (and does that include the glassware on the bottom?). You'll have 2 of the anchors, yes, but I wouldn't want to load 90+ lbs on toggle bolts - I just wouldn't trust the drywall that much. Any un-level installation or uneven loading would skew those numbers, so I'd stick to keeping it half full or less (hey, drink more wine!).
As an alternative, could you mount a piece of 1/2" plywood to the wall, hitting studs, and then mount the wine rack to the plywood? If your wall doesn't have studs, what does it have?