First things first, if you have a pet spider that you feel is not well, you should look for a specialist exotic pet veterinarian. However, as for the question of being able to take a pet spider to the vets, it is unlikely that you are going to be able to take them to your local vets and expect help. People are still people, and many people have fears of spiders, they are, in fact, one of the largest phobias in the world. However, if you do not own one already, you should certainly ring around your local vet centers and see if you will have any luck taking them there.
If you bought your pet spider from a local pet shop, you may have more luck going back to them for advice. They are likely well versed with pet spiders and have experience finding help for them.