Is it possible to create blood cells that produce their own oxygen?

No, from what would they produce it? It would take more energy to break down water than the energy gained from the Oxygen.

No, even if you put it into practice in a laboratory. It is not ethical, and the successful rate is low.
Many people failed. So I say not.

Hypothetically, yes. That would be an interesting secondary endosymbiosis. The erythrocytes would need to have endosymbiosis with an alga.

Why chloroplasts alone wouldn't work: chloroplasts in plants have surrendered too many of their genes to the nucleus. Us animals don't have the nuclear genes needed to keep plant chloroplasts alive and active.

The oxygen would come from water, just as it does in other oxygenic photosynthetic organisms.

Why this wouldn't work very well: We don't have a big enough surface area exposed to light to produce a useful amount of oxygen in a useful amount of time.

Keep breathing, my friend.

Sure, you just need to add chloroplasts to hemoglobin, then expose the blood to light. But might I suggest trying this with skin cells instead of blood cells? The whole "exposing them to light" part is easier to accomplish with skin cells.

And where does this oxygen come from?

They should be photosynthetic. In photosynthesis, oxygen is produced from splitting water, but I do not think so

Which blood cells do you think should carry out this task? Instead of them doing it why not have all cells simply produce their own. Can you suggest the mechanism whereby cells take the element oxygen and produce dioxygen molecules. From where are they going to acquire the oxygen?

Theoretically possible in the future, they called reciprocytes.

What would they use as a source material?

I doubt it. If it were our lungs and associated items would become redundant. We could also swim underwater indefinitely. Pipe dream.