No, this is not possible. There are a few reasons for that, but most important are that the only thing a mosquito injects is its own saliva, while the blood is sucked into the stomach where it is digested.
To be able to infect other people HIV would need to be able to leave the gut intact and then also be able to replicate in the mosquitos which it cannot do, due to the missing of the CD4 antigen on the surface of the insect cells. These are needed as a surface receptor for the virus to bind and enter the cells. This is also true for other blood sucking insects like bed bugs or fleas.
Other pathogens can do this, examples would be Yellow fever or Malaria. In Yellow fever the virus first infects epithelial cells of the gut, then enters the blood system of the insect to finally end up in the salivary glands, where the virus is injected together with the saliva into the biten person. In Malaria the pathogen is also able to leave the gut region and mature in the salivary glands.
HIV can only be transmitted through blood (either through direct transmission, operations etc.), through semen (cum), pre-seminal fluid (pre-cum), rectal fluids, vaginal fluids, and breast milk. See reference 3.
- Why Mosquitoes cannot transmit AIDS
- Can we get AIDS from mosquito bites?
- HIV Transmission Risk: A Summary of Evidence