As-usual, be cautions of many of today's wildfire-like social media - or even traditional media - pass-alongs. These are low-hanging fruit that often prey on both our worries about the current social situation as well as tend to sensationalize:
Hand sanitizer is definitely flammable ... but enough to start a car fire?
www.snopes.com
With that said......
Is it possible?
Yes.
Is it likely?
In my personal experience, no.
Most health-care professionals I know (and I know a lot - my wife and I both went through medical school in the late-90s/early-oughts

; she remains a licensed pediatrician, and I'm currently a basic-science researcher, so our friend-group practically since our undergrad days are all in the biomedical sciences) have had bottles of Purell sitting in their vehicles in-perpetuity even well before SARS-CoV-2 and COVID-19 became household names. None of our friends or even acquaintances have ever reported such a problem, and we've certainly had various sized bottles in our personal vehicles for, well, going on decades now, in all kinds of environments (including with the vehicle out in open parking lots, for literally days, in South Carolina, Georgia, Florida, and Texas sun, smack in the middle of summer during vacation seasons).
It's like those tales - and physical demonstrations - of water-bottles or even external concerns (such as mirrored surfaces from, say, a building across the parking lot:
This London skyscraper can melt cars and set buildings on fire and
London's Burning: How a Skyscraper Melts Cars) forming lenses and setting the vehicle's interior materials aflame.
Can it happen.
It certainly can.
But it's not going to be an everyday occurrence.
The more practical worry is that in today's pathogen-conscious environment, we're eager to get our hands - and that of our kiddos' - sanitized, and it's worth remembering that everything inside the car will reach excessive temperatures in the hot summer (regardless whether the car is parked in direct sunlight or not, although shading can delay/mitigate more rapid temperature rises). That gel-like sanitizer that you squeeze out into your or your child's hands can cause quite a bit of discomfort, or even potentially a first-degree burn (translation: like a sunburn). Those of us who are from hotter climates either to the south or the west of the country likely know this already, but for those of us in the north who are not used to having bottles of sanitizer in the car, it's worth remembering now that the sunnier/hotter months are here!
While there have been no officially documented report of static discharge igniting hand-sanitizer in-use (this from the WHO, so it's international, not just Stateside or even North-American, and is current at least as of a month ago, when I last checked), it is nevertheless prudent to remember that particularly the off-brand and newer entries to-market are even higher in alcohol content than the Go-Jo made Purell that's come to define the breed. If you are a smoker or are otherwise using the sanitizer on yourself or your children near open flames (i.e. cooking fire, be it indoors or out), be cognizant of the increased potential to cause actual second/third-degree burns in such a context.
Finally, remember how these gels/liquids work: that the alcohol percentage is a big chunk of that equation -
How does hand sanitizer work? - and that this can be negatively impacted by evaporation, which can be hastened with high-temperature storage (and yes, this even applies to that little bottle you've put in your pants pocket - but since that bottle will most likely be used rather quickly and refilled/replaced, it's not as much of a concern as that giant bottle you're keeping in your car's center console).