Learn Hardcore Polish: Jutro mam szczepienie, więc pytam, czy zastrzyk będzie bolał. - Tomorrow I have a vaccination, so I ask whether the injection will hurt.

Breakdown of Jutro mam szczepienie, więc pytam, czy zastrzyk będzie bolał.

Why does mam szczepienie literally use mam for I have instead of something like I will get vaccinated?

This is a very natural Polish way to say it.

Mam szczepienie literally means I have a vaccination, but in real usage it often means:

Polish often uses mieć for appointments, scheduled events, and things you are due to undergo, for example:

So Jutro mam szczepienie is idiomatic and natural.

What is the difference between szczepienie and zastrzyk in this sentence?

They are related, but not the same.

szczepienie = vaccination, immunization, or the vaccination procedure as a whole
zastrzyk = injection, shot, the actual needle injection

So the sentence first talks about the whole event:

Then it asks about the specific part that might hurt:

That is why both words appear.