Yes, mosquitoes pollinate flowers. The normal food of adult mosquitoes is nectar from plants.
Most people think of mosquitoes only as insects that suck blood, but they have another niche in the ecosystem–they pollinate flowers. Male mosquitoes never bite, and the females need the protein in blood only to produce eggs, so the normal food of adult mosquitoes is actually nectar from plants.
Though they don’t gather pollen like bees, they fly from flower to flower to feed, and along the way, they carry pollen from one blossom to the other.
In most cases, mosquitoes are just one of many insects that plants use for pollination, so even if mosquitoes were all eradicated by humans, plants would still survive. Some species are more vulnerable than others. One common plant, that isn’t at any risk, though is…
Goldenrod uses mosquitoes, as well as other insects, for pollination. You might think it appropriate that an annoying insect would help pollinate a flower notorious for causing hay-fever, but scientists believe that goldenrod really isn’t responsible for fall allergies, compared to other less noticeable plants that bloom at the same time, but which have smaller grains of pollen that blow in the wind. 
The sticky grains of goldenrod pollen adhere to mosquitoes and other insects as they forage over the blooms.
“Twenty chapters on twenty ‘bad bugs’ that have taught us more about our world, including some surprising things we maybe didn’t want to know.”
But there’s no need to worry about goldenrod going extinct. Enough other insects feed on goldenrod nectar that getting rid of mosquitoes wouldn’t have any effect.
Mosquitoes also act as pollinators for grasses and a few other flowering plants. Their role in spreading pollen hasn’t been studied nearly as much as their role in drinking blood and spreading diseases. But the niche they’re best known for is pollination orchids, including the blunt-leaved bog orchid,(Habenaria obtusata, also called Platanthera obtusata), and other rare Arctic bog orchids.
John Smith Dexter was the first to report that mosquitoes pollinated orchids, in 1913.  A researcher, Ada K. Dietz, noticed a mosquito in Michigan that had two yellow masses of something stuck to its head. Dexter went to investigate and found other similar mosquitoes. “In a few minutes I had caught a half dozen or more, all of them females, bearing the yellow masses,” he reported. He discovered the pollen was from the blunt-leaved bog orchid that was blooming nearby. The mosquitoes who had been feeding on the orchid had one, two or three bits of pollen stuck to their eyes. One had four grains.
Dexter put some of the orchids and some mosquitoes in a glass aquarium and within a few days, most of the mosquitoes also had pollen stuck to their eyes in the same way. They picked them up from feeding on the orchid blooms, and eventually would transfer them to orchids when they fed on them and the pollen fell off.
In the years since Dexter reported his discovery, scientists have found mosquitoes pollinating other varieties of orchids too, particularly in the northern U.S., Canada and Alaska. 
The fact that orchids use mosquitoes for pollination is just one more surprise in the wild and crazy history of these rare plants.
Mosquitoes aren’t the only insects which can pollinate the orchids, however. Moths also pollinate the bog orchid that Dexter studied.  A few rare Arctic orchids, growing in ecological niches where insects are scarce, may be more dependent on mosquitoes for pollination, though they may also be partially self-pollinating.
Other plants and animals use mosquitoes for food or pollination, but if mosquitoes were entirely eliminating by humans, rare arctic orchids would probably suffer the most, since their ecological niche is more dependent on mosquitoes, and many of them grow where there are few other insect options.
Other Questions & Answers
- Do mosquitoes sleep?
- How long do mosquitoes live?
- What blood type do mosquitoes prefer?
- What do mosquitoes eat?
- When is mosquito season?
- Where do mosquitoes go in the cold?
- Why do mosquitoes buzz?
- Why do mosquitoes exist?
 Goldenrod Biology.
 Mosquitoes Pollinating Orchids, Science magazine, June 6, 1913. John Smith Dexter.
 Orchid Pollination by Aedes Mosquitoes in Alaska.
 An Atlas of Orchid Pollination by Nelis A. Cingel.
Public domain photos of goldenrod by Pollinator and orchid by Jason Hollinger, Wikimedia Commons.