Definitely not normal Bacteria:
I'm going to narrow my focus on how a bacteria could replicate the iconic part of blood that makes it "blood." The red blood cell. Actual whole blood is a complex mixture that would require a matrix of replicating pluripotent eukaryotic stem cells and other additional factors a little more complex than can be dealt with here.
By definition, bacteria are prokaryotic organisms. They are genetically simple (yes, that's a loaded term) and don't have a nucleus. Their existence is driven by replication. Most (with a few exceptions) are tiny compared to eukaryotic cells. So how would a simple prokaryotic organism make a complex eukaryotic cell?
This is not as bad of a question as it seems on first glance. Red blood cells in humans don't have a nucleus. There are very few cells that don't. They can't replicate on their own and derive from nucleated red cells that have their nuclei removed. Most bacteria other than Mycobacteria have cell walls to cope with osmotic pressure from their environment. Mycobacteria are often intracellular parasites and tend to have even simpler genomes than most bacteria because they are depending on their hosts for biosynthesis. So how does a tiny, super-simple cell become something immense and complex? There are also some big bacteria but these tend to be highly specialized to exotic environments. For something as small and simple as most bacteria to make a red blood cell, they would need to work as a team. Many bacteria can organize themselves into multicellular structures that seem to carry out more complex tasks, but are generally not considered truly multicellular.
The biggest barrier in all this to having bacteria making blood is energy. The bacteria would not have the genome to make blood, or the manufacturing capacity to do so, but most critically there is no energetic advantage to making these huge, expensive, non-replicating red cells. Even if the bacteria could, they would need to be kept in a bioreactor of some sort with massive inputs of nutrients. Scientists certainly make genetically engineered bacteria that can produce large amounts of simple excreted materials, but the advantage is that the organism survives because the scientist "rewards" the species with survival for producing what they want.
If you need to go as far as having a bioreactor to produce red blood cells, you're better off growing pluripotent stem cells in culture, then forcing them to mature into red blood cells through maturation signals. It wouldn't (won't?) surprise me when scientists start using just this approach to produce blood outside the body for transfusion purposes. Specialized cells lacking surface antigens to minimize antibody interactions will probably replace blood donations eventually.
So you can have animals making blood (Maybe even unicellular ones, but at that point they're essentially stem cells). You can have bacteria making something that looks like blood but isn't. You can have bioreactors that crank out the cells for blood, and you could probably have bacteria make a lot of the plasma components for blood. I don't see how the bacteria could or would make blood on their own.