Best answer would be to call Chase directly. With the right credit score and income, Chase approves immediately at dealerships all of the time. I purchased a Subaru and the captive lender for Subaru is Chase bank. I drove home from the dealership with the car.
If there are any issues with the application there could be more time required for approval. Example, lower credit score, unable to easily verify income, short credit history.
From your post I am assuming you applied through Chase and not the dealer? If that is the case, call Chase and explain your situation, they should be able to tell you when they will fund the loan.
If you financed through the dealer, give them a call and see what they say for a timeline.
When I purchased my Subaru it took 19 days from the time I drove my car home until the loan showed up on my Chase.com account.
Let us know how things work out, good luck. I hope you are getting a good deal on the rental car. I have had pretty good luck with Enterprise having good weekly rates.
