You need to look at the rules for your jurisdiction to see what is allowed, but in no case that I can think of would the contractor themselves pull the permit and give it to you and let you do the work, or have someone else do it. You might find a contractor that would agree to start the process but delay the start of the work and further payments. Are you just trying to spread out payments, or are you just trying to get the process started and be ready? Why are you looking to do this? Why not pull the permits yourself if that is allowed? If not allowed because the permit has to be issued to a licensed contractor, then obviously the permit stays with that contractor.
In my city, you have to be a contractor licensed in the city to pull a permit and do the work. The permit is tied to the contractor. As the owner, you can pull your own permits, but you sign a form stating that you are doing 100% of the work yourself. If you were paying anyone to do any work (like hourly labor), you would need to apply and become a licensed contractor and have insurance, etc. If you were hiring a contractor, they need to pull the permit. I believe it was also possible for the owner to pull various permits and have someone else do the work on a complete project, but you had to have the name of the contractor and that contractor had to be licensed in the city.