Yes, it is important that you go to the same provider/clinic for your first and second vaccinations. Second doses are allocated to providers based on the initial dose count they received. If too many people go to different providers for their second dose, it makes it very difficult to keep track of the proper doses needed for each second dose clinic.
If you are unable to get a second dose with your original provider, the University of Washington is offering a second dose clinic regardless of where the first dose was given. They have both Pfizer and Moderna.
You can access this service by calling UW at 206-520-8700, press 3, then press * and hold. Then will register you or locate your existing record. Appointments are available at Montlake, Harborview, or Northwest.