At least in the US it is normal practice for pets to leave with all relevant vaccines (as well as neutering and routine medical care like deworming). All of the shelters in my area do this, and two are part of national organizations. @mhwombat notes in a comment that this is also true in Ireland and the UK.
More specifically, your new pet will have received all vaccines that are due. If you adopt a young animal there may be vaccines that can't be given yet; for example, one of the vaccines that kittens get has to be given in three doses spread over several weeks. If you were to adopt a kitten before that course is complete (and I don't know if a shelter would release the kitten in that case, but maybe), you would have to complete the cycle. I've been told that puppies have vaccines due around 5-6 months. Other animals may also have post-adoption first-year vaccine needs, and you would be responsible for those.
The shelter will go over all known medical history at the time of adoption, including what vaccines the shelter administered and when. You'll need to pass those records on to your vet so that future vaccines are given on schedule. If there were to be anything that will be due soon, that the shelter didn't administer, I would expect them to tell you that.
Finally, you asked if there is a chance you will have to pay for these yourself. Having a pet involves a financial commitment; if the cost of a vaccine makes the difference for you, please think carefully about whether you'll be able to care for the pet if it gets sick or is in an accident. Pets are wonderful additions to a household, but please make sure you can afford them.