Absolutely not. Their proboscis is designed to pierce skin not shell. A mosquito can land on, but not bite into - let alone bite through - the shell.That's an interesting fact about the mosquitos on the carapace/plastron.
It's dumb question time.
Do they actually get blood from the carapace or plastrons?
They're attracted to warm blooded mammals by the carbon dioxide they exhale. As torts are cold blooded reptiles and further defended by scales, mosquitoes are not a problem to them.
And that photo does not show a tort being bitten through its shell.