I wouldn't want a transfusion from somebody infected with this bug.
The people who show heart valve damage or get diarrhea because their GI tract is ravaged by the virus didn't have their hearts or GI tract exposed to air-borne droplets contain SARS-CoV-2 virions. Once a cell produces the virus and is lysed, the virus can travel in the bloodstream. It must in order to produce the effects we see.
It's not blood borne because we're not around others' blood all that much. I'd suspect it could be blood borne. I'd hope that they checked the blood supply for blood drawn in January-March.
That gets back to the OP's question, which doesn't just involve stinging but the blood meal that female mosquitoes need.
I think the answer is almost certainly that mosquitoes can't pass the virus. The mosquito may bite different people to top off her tank with blood, injecting an anticoagulant each time, but the virus won't infect the mosquito and I don't see why it would travel from stomach to whatever organ produces the anticoagulant and show up in that fluid. It's not like she injects the blood from the first "donor".
And in any event it would be a small amount--the blood won't have that high a concentration, and if some stray virion manages to get across the stomach barrier into that fluid it's not enough to produce an infection.