Most vaccines are injected into the deltoid muscle to optimise the immunogenicity of the vaccine and to minimise adverse reactions at the injection site. The intramuscular route is by far the most effective, an alternative to the deltoid muscle is the anterolateral aspect of the thigh.

Sugar cube polio vaccine when I was a kid.

Most vaccines would get destroyed by your stomach juices. I have to inject insulin for the same reason.

It depends on best absorption and where the best absorption is. I have to inject B-12 once a week. Most people can absorb that through the digestive tract. My body won't. Injection is a sure fire way to get it into the body, other methods aren't as guaranteed. Depending on what the person has eaten, etc.

Polio and other organisms that can infect through the digestive tract are good candidates for oral vaccines. Respiratory viruses and others that infect through blood and mucus are too fragile for that route. There are nasal flu vaccines. Your respiratory and digestive systems are evolved to protect you; the injection bypasses much of your passive immune defense.

Good question! I don't know the answer. I remember taking my polio on a sugar cube.

It's just a guess, but...
I should think that the most effective way to get something into the bloodstream (where the white blood cells are) would be by venipuncture. But this is more difficult, so the second-best way is by random injection somewhere on your body.

My kiddo did have a couple of innoculations by nasal spray. This was probably less effective, but still good enough.

Again, just a guess. I'm better at drawing blood than I am at putting stuff into it.