Interesting. I'm going to say zero chance of infection for a couple of reasons,
1.
Zombie infection may not be blood-bourne
From every movie, videogame and TV series that I've seen about Zombies, people get infected from being bitten by a Zombie. This implies a blood-bourne infection (i.e. you get the infection from contact with a Zombie's blood). In general, infectious diseases that are blood bourne are also transmitted by saliva and visa-versa, but I'm not sure that is necessarily true for Zombie-ism. Zombie infection seems to be only spread by biting (similar to rabies), suggesting that the infection is only in the saliva of Zombies. There are many examples in movies, videogames and TV, where even large exposure to the blood of a zombie does not result in disease transmission (most notably in the excellent Walking Dead episode where they covered themselves in Zombie guts to blend in the the other Zombies in order to escape). Suggesting that,
2.
Zombie infections are hard to catch
But say you could catch a Zombie infection from the blood of a Zombie, it seems that this infection is really hard to catch. Take two well-studied diseases, Hepatitis B and HIV. It is really easy to catch Hepatitis B. Even small blood exposures can result in infection. But, Zombie infection seems more similar to HIV, which is relatively hard to acquire. Unless you are exposed to significant amounts of blood (splashed in eye, transfusions, etc), your chance of catching HIV is low. In this respect, Zombie infection seems more similar to a hard-to-acquire infection, such as HIV.
And we know from the HIV epidemic that you really can't get HIV infection from mosquitos. I believe that all of those cases of mosquito-transmission have been debunked (shocking that people lie about thier sexual history or IV drug use!). Which leads me to,
3.
Mosquitos generally only spread infections of organisms that use mosquitos as part of their life cycle.
This last one is tougher to explain. Mosquitos (or other blood sucking parasites) spread lots of infections, but the infecting organism needs to be able to live and survive in the mosquito, otherwise, it can not reproduce and spread to another animal. See the above figure for details
Clearly, I have spent way too much time on this today!