Learn German: Wir lassen den Hund beim Tierarzt impfen. - We are having the dog vaccinated at the vet.

This is a major difference between English and German. English uses a passive structure ("having the dog vaccinated"), but German causative sentences pair lassen with the active infinitive at the end. Literally, it is closer to "We let [someone] vaccinate the dog." Always use the plain infinitive with lassen.

How do we know lassen means "having something done" here, rather than "to leave"?

You can tell by the second verb! When lassen stands alone, it usually means "to leave" (like leaving your keys at home). But when it is paired with an infinitive verb at the end of the sentence (like impfen here), it acts as a causative verb, meaning you are arranging for someone else to do the action.

Why is impfen placed at the very end of the sentence?

lassen behaves structurally just like a modal verb (such as können or müssen). The conjugated verb (lassen) takes the second position in the clause, and it pushes the second verb (impfen) to the very end in its basic infinitive form.

Why does the sentence use beim Tierarzt instead of just bei Tierarzt?