No one seems to have mentioned pumpkin yet.
That’s because no one misses them when they are gone. Even while they are around, there is a spice named after them but it doesn’t taste anything like pumpkins.
Kumquats are seasonal, sadly. Nov-Feb, usually.
As are pomegranates, Sept-Feb in the north, Mar-May in the southern hemi.
I feel unloved. 
Fiddleheads are in some of the stores around Boston, but more common in NH and ME when in season. I doubt they’re in much demand except in a few areas of the country to bother with shipping them around. They’re a specialty item, for sure.
Ramps are much more rare. They are only in specialty shops and I don’t think they travel well. The supply is very limited so it’s hard to imagine them in a major supermarket in any volume.
There is this company that grows cotton candy grapes (fucking amazing) and they are only available for one month (aug-sept) each year.
I see stone fruits, asparagus, apples, citrus, and other “seasonal” items in my local supermarket year round. In the off-season they are from the southern hemisphere, quite expensive, and not very good. There are always apples and grapes.
Pumpkins are weird – or rather the market for them is weird. You can sell them right up through October 31st. On November 1st you may as well plow them back into the field.
What is weird about this is that it doesn’t only apply to jack-o-lantern pumpkins, for which it makes sense; most of them are too tasteless, watery, and/or stringy to bother eating. It also applies to pie pumpkins. You’d think there’d be another surge of people buying those right before Thanksgiving; but there generally isn’t. I do usually keep stubbornly trying to sell them, if I’m doing winter markets; and once in a while somebody buys one or two; but mostly they don’t.
(This is, I realize, a very USA centric post.)
Speaking as somebody who prefers pie pumpkins even for making jack-o-lanterns with, thank you! I think canned pumpkin is boring and cutting up a pumpkin for decorative purposes only to let it rot is wasteful. Also I think there’s nothing to beat fresh-roasted pumpkin seeds.
The way to do jack-o-lanterns right is to get a good pie pumpkin, carefully extract and wash the seeds, and carve and display your jack-o-lantern as preferred, saving any sizable chunks of flesh. As soon as trick-or-treating is over, bring Jack back into the kitchen, scrape off any candle wax or scorch marks from inside him, split his head in two from ear to ear, put the halves convex side up on a baking pan, and roast him till his smiling face has gradually slumped into a soft mass. Mwa-ha-ha-ha.
You can roast the seeds at the same time. When the pumpkin’s cool, scrape out all the flesh, puree it well, and freeze for pumpkin pie/bread/cheesecake that’s much tastier than anything you’ll make with a can of One-Pie.
Lychees and longans were the first fruits to come to mind. I don’t think I’ve ever seen them in the winter.
Definitely shell peas.
I’ve only ever seen russet apples in the fall.
Hatch green chiles are seasonal.
Prickly pear fruit is seasonal as well.
In my experience, all American fruits and vegetables taste like they were carved out of balsa wood.
I see a lot of people are citing the idea that only ‘good’ examples of name a fruit are available at certain times of year - well, duh, that’s what seasonality is about. If I read the OP correctly, they’re asking what just isn’t available at all off-season.
I know the question was about US supermarkets, but the British supermarket offering is similar - the vast majority of stuff is available all year round, even if it’s rubbish, flown in from Brazil, South Africa and wherever. So whilst we know that strawberries are only really good May-July, we can still get them in January.
Anyway, I used to do a lot of branding/ad work for a major supermarket, so have a pretty clear idea of what is just not available out of season, mainly because I needed to shoot fruit six months ahead of the season. And it’s basically fresh cranberries and chestnuts. Cherries can be iffy to find too. Pretty much everything else is freely available, even if it tastes like cardboard.
Oh, jersey royal potatoes. But I guess most people here won’t know what they are (food of gods).
Really, it’s all varieties of melons that are only available in summer as fresh fruit, and they’re not really good until the second half of June.
For example you can get pre-cut melon containers in January, but not the actual watermelons, cantaloupes or honeydews themselves.
Certain varieties of citrus seem to be seasonal as well- stuff like Meyer lemons, some types of grapefruit, etc… are only available in the winter (when citrus ripens). I mean, I suppose I could go hunt down Meyer lemons at the local gourmet store in July, but they’re not available at the local mega-marts except in November through about March (if that long).
I’ve never seen Blood Oranges available off-season.
Mineola tangelos seem hit & miss, and not linked to any “season” as such. Like a sine wave: For a couple or 3 months they’re all around, then a major dearth of them for a couple of months, then a couple of months of plentiful supply, then a couple of months…
My favorite orange of them all. Sweet, juicy, very easy to peel. Never knew how good Long Island was for growing oranges. 
You beat me to it! They are absolutely delicious. There’s a restaurant called Suzanne’s in Ojai that makes a Pixie Cosmo. Yummy with their crab salad on a hot summer day sitting on the patio.
In Hawaii, I see whole cantaloupes and honeydew melons year round. I think those mini Thai watermelons are year round too. I suspect it’s because these are always imported, never local so we don’t really get the “good stuff” anyway.
If something is out of season in America, it will be back in season in Chile before too long and–I’m not going to guess a number, but it’s probably more than 30%–a lot of our produce comes from Chile.
The seeds from Jack-o-lantern pumpkins are usually fine. I always separate them and roast them as part of the Jack-o-lantern event. They rarely last the week.
(I don’t care for pumpkin flesh enough to bother cooking my own. I suppose it’s possible I’ve just never had good pumpkin. And I do like butternut and acorn squash. But I’m content to carve a large, flavorless decoration and toss it in the compost heap when it starts to rot.)