You should take your new pet to your vet for a healthy pet check up, regardless of where you got it. Many shelters (all that I work with) will have vet checked your pet as part of the intake and maintenance of the animal as part of their program. All the shelters in my area have vets on staff as full time positions, in my opinion shelter pets are more likely to have seen a vet more recently than those purchased at a store, or found online.
It is important that you take any new pet to see your veterinarian for a well pet check-up as soon as practical. A quick Google search found many veterinary practices in my area offer free new pet checks.
Why is this important? There are two key points.
When the vet sees your pet in a healthy state, they will have a base line to compare to if anything develops in the future.
If you have a medical emergency during off hours, many neighborhood vets will not see animals they have not worked with before.
While there is always some chance that the new pet vet check will find something, and allow it to be addressed in a timely manor. The most important thing is establishing a relationship between your pet and your vet.